1. Abridged Problem Statement

There are `n` ropes attached at points `(xi, 0)` on the horizontal axis. Their other ends are tied to a point-like ball. Rope `i` has fixed length `li`, so the ball must always lie inside or on the disk centered at `(xi, 0)` with radius `li`.

Initially all ropes are present and the ball is at rest. Then `n-1` ropes are cut in a given order. The first cut happens at time `0`; each next cut happens as soon as the ball becomes stationary after the previous cut. If cutting a rope does not move the ball, the next cut is immediate.

The ball moves in a perfectly viscous medium: at every moment it moves in the feasible direction that decreases potential energy fastest, i.e. as downward as possible, with constant speed `1`.

For each query time `t`, output the ball position.

Constraints: `2 ≤ n ≤ 20`, `1 ≤ m ≤ 1000`.

2. Detailed Editorial

Geometry model

For every remaining rope `i`, the ball must satisfy

(x-xi)^2 + y^2 <= li^2

because the distance from the attachment point `(xi, 0)` to the ball cannot exceed the rope length.

So every rope corresponds to a closed disk. If the active rope set is `S`, then all feasible ball positions are in the intersection of disks D_i for i in S, where `Di` is the disk of rope `i`.

Since gravity pulls downward, a static position is the feasible point with minimum `y`.

Finding the equilibrium point

For a fixed active set `S`, we need the lowest point in the intersection of disks.

The optimum must lie on the boundary of at least one disk. In 2D, it is enough to check:

1. The bottom point of each active circle: `(xi, -li)`
2. Every intersection point of two active circles.

For each candidate point, check whether it lies inside all active disks. Among valid candidates, choose the one with minimum `y`.

Since `n ≤ 20`, this brute-force candidate enumeration is easily fast enough.

Motion after a cut

After a rope is cut, the feasible region becomes larger. The current position is still feasible, but it may no longer be the lowest feasible point. The ball moves until it reaches the new equilibrium.

At any point `p`, define the tight ropes as ropes whose distance from `p` is exactly equal to their length. These are the ropes currently stretched.

For a tight rope `i`, its outward unit normal is `n_i = (p-(xi,0)) / li`.

A feasible velocity direction `d` must not move outside a tight disk: `d . n_i <= 0`.

Gravity direction is `g = (0,-1)`.

The ball chooses the feasible direction closest to gravity, i.e. the projection of `g` onto the feasible tangent cone.

There are several cases:

Case 1: Gravity is feasible

If `g . n_i <= 0` for all tight ropes, the ball falls straight down. This produces a vertical segment until it hits another circle boundary, or until it reaches the target equilibrium.

Case 2: Gravity is blocked by one or more ropes

Then the ball slides along some tight circle. For each tight rope `k`, project gravity onto the tangent line of circle `k`:

`d_k = g - (g . n_k) n_k`

Normalize it, keep only directions that go downward and remain feasible for all other tight ropes.

The best such direction determines which circle arc the ball follows.

Case 3: No downward feasible direction exists

The ball is already at equilibrium.

Path representation

The whole path between two equilibria consists of simple pieces:

1. Vertical segments.
2. Circular arcs along one rope.

For each phase after a cut, we store:

- start time,
- start point,
- end equilibrium point,
- list of path segments,
- total length.

Since speed is `1`, elapsed time equals path length.

Processing queries

After preprocessing all cuts, we have a sequence of motion phases.

For a query time `t`:

1. Find the first phase whose time interval contains `t`.
2. Let `s = t - phase.start_time`.
3. Walk through that phase's segments until distance `s` is reached.
4. Interpolate inside the current vertical segment or arc.
5. If `t` is after all phases, output the final static position.

Complexity

`n ≤ 20`, so even repeated pairwise circle intersection checks are cheap.

Equilibrium computation is roughly `O(n^3)` because there are `O(n^2)` candidates and each is checked against `O(n)` disks.

The trajectory tracing is event-based and small for these constraints.

3. C++ Solution

```cpp
#include <bits/stdc++.h>

using namespace std;

template<typename T1, typename T2>
ostream& operator<<(ostream& out, const pair<T1, T2>& x) {
    return out << x.first << ' ' << x.second;
}

template<typename T1, typename T2>
istream& operator>>(istream& in, pair<T1, T2>& x) {
    return in >> x.first >> x.second;
}

template<typename T>
istream& operator>>(istream& in, vector<T>& a) {
    for(auto& x: a) {
        in >> x;
    }
    return in;
};

template<typename T>
ostream& operator<<(ostream& out, const vector<T>& a) {
    for(auto x: a) {
        out << x << ' ';
    }
    return out;
};

using coord_t = long double;

struct Point {
    static constexpr coord_t eps = 1e-12L;
    static inline const coord_t PI = acosl((coord_t)-1.0L);

    coord_t x, y;
    Point(coord_t x = 0, coord_t y = 0) : x(x), y(y) {}

    Point operator+(const Point& p) const { return Point(x + p.x, y + p.y); }
    Point operator-(const Point& p) const { return Point(x - p.x, y - p.y); }
    Point operator*(coord_t c) const { return Point(x * c, y * c); }
    Point operator/(coord_t c) const { return Point(x / c, y / c); }

    coord_t dot(const Point& p) const { return x * p.x + y * p.y; }
    coord_t norm2() const { return x * x + y * y; }
    coord_t norm() const { return sqrtl(norm2()); }

    friend ostream& operator<<(ostream& os, const Point& p) {
        return os << p.x << ' ' << p.y;
    }
    friend istream& operator>>(istream& is, Point& p) {
        return is >> p.x >> p.y;
    }
};

int n, m;
vector<coord_t> xs, ls;
vector<int> cut_seq;
vector<coord_t> ts;

bool inside_disks(const Point& p, const vector<int>& S, coord_t tol = 1e-7L) {
    for(int i: S) {
        coord_t dx = p.x - xs[i], dy = p.y;
        if(dx * dx + dy * dy > ls[i] * ls[i] + tol) {
            return false;
        }
    }

    return true;
}

vector<Point> circle_pair_inters(int i, int j) {
    coord_t dx = xs[j] - xs[i];
    coord_t d = fabsl(dx);
    if(d < 1e-14L) {
        return {};
    }

    coord_t r1 = ls[i], r2 = ls[j];
    if(d > r1 + r2 + 1e-12L || d < fabsl(r1 - r2) - 1e-12L) {
        return {};
    }

    coord_t a = (r1 * r1 - r2 * r2 + d * d) / (2 * d);
    coord_t h2 = r1 * r1 - a * a;
    if(h2 < 0) {
        h2 = 0;
    }

    coord_t h = sqrtl(h2);
    coord_t sgn = (xs[j] > xs[i]) ? 1.0L : -1.0L;
    coord_t mx = xs[i] + a * sgn;
    return {Point(mx, h), Point(mx, -h)};
}

Point equilibrium(const vector<int>& S) {
    if(S.empty()) {
        return Point(0, -1e18L);
    }

    vector<Point> cands;
    for(int i: S) {
        cands.push_back(Point(xs[i], -ls[i]));
    }

    for(size_t a = 0; a < S.size(); ++a) {
        for(size_t b = a + 1; b < S.size(); ++b) {
            for(auto& q: circle_pair_inters(S[a], S[b])) {
                cands.push_back(q);
            }
        }
    }

    Point best(0, 1e18L);
    for(auto& c: cands) {
        if(inside_disks(c, S) && c.y < best.y) {
            best = c;
        }
    }

    return best;
}

vector<int> tight_set(const Point& p, const vector<int>& S) {
    vector<int> tight;
    for(int i: S) {
        coord_t dx = p.x - xs[i], dy = p.y;
        coord_t d = sqrtl(dx * dx + dy * dy);
        if(fabsl(d - ls[i]) < 1e-6L) {
            tight.push_back(i);
        }
    }

    return tight;
}

struct DirResult {
    Point dir;
    int arc_circle;
};

DirResult descent_dir(const Point& p, const vector<int>& tight) {
    Point g(0, -1);
    if(tight.empty()) {
        return {g, -1};
    }

    vector<Point> normals;
    for(int i: tight) {
        Point r(p.x - xs[i], p.y);
        coord_t rn = r.norm();
        if(rn < 1e-14L) {
            normals.push_back(Point(0, -1));
        } else {
            normals.push_back(r / rn);
        }
    }

    bool g_ok = true;
    for(auto& nk: normals) {
        if(g.dot(nk) > 1e-9L) {
            g_ok = false;
            break;
        }
    }
    if(g_ok) {
        int tangent_circle = -1;
        for(size_t k = 0; k < tight.size(); ++k) {
            if(fabsl(g.dot(normals[k])) < 1e-9L) {
                tangent_circle = tight[k];
                break;
            }
        }

        return {g, tangent_circle};
    }

    DirResult best;
    best.dir = Point(0, 1);
    best.arc_circle = -1;
    bool found = false;
    for(size_t k = 0; k < tight.size(); ++k) {
        const Point& nk = normals[k];
        coord_t gn = g.dot(nk);
        Point d(-gn * nk.x, -1 - gn * nk.y);
        coord_t dn = d.norm();
        if(dn < 1e-14L) {
            continue;
        }

        d = d / dn;
        if(d.y >= -1e-12L) {
            continue;
        }

        bool ok = true;
        for(size_t kk = 0; kk < tight.size(); ++kk) {
            if(kk == k) {
                continue;
            }
            if(d.dot(normals[kk]) > 1e-9L) {
                ok = false;
                break;
            }
        }
        if(!ok) {
            continue;
        }

        if(!found || d.y < best.dir.y) {
            best.dir = d;
            best.arc_circle = tight[k];
            found = true;
        }
    }

    if(!found) {
        return {Point(0, 0), -1};
    }

    return best;
}

struct Segment {
    bool is_vert;
    coord_t x, y0, y1;
    coord_t cx, r, theta0, theta1;
    coord_t len;
};

struct Phase {
    coord_t start_t;
    coord_t total_len;
    Point start_p, end_p;
    vector<Segment> segs;
};

coord_t fwd_angle(coord_t from, coord_t to, coord_t s) {
    coord_t two_pi = 2 * Point::PI;
    coord_t dt = s * (to - from);
    dt = fmodl(dt, two_pi);
    if(dt < 0) {
        dt += two_pi;
    }
    return dt;
}

vector<Segment> trace_path(Point start, Point target, const vector<int>& S) {
    vector<Segment> segs;
    Point cur = start;
    coord_t two_pi = 2 * Point::PI;
    for(int iter = 0; iter < 20000; ++iter) {
        if((cur - target).norm() < 1e-9L) {
            break;
        }

        vector<int> tight = tight_set(cur, S);
        DirResult dr = descent_dir(cur, tight);
        if(dr.dir.norm() < 1e-9L) {
            break;
        }

        if(dr.arc_circle == -1) {
            coord_t y_end = -1e18L;
            for(int i: S) {
                bool is_t = false;
                for(int t: tight) {
                    if(t == i) {
                        is_t = true;
                        break;
                    }
                }
                if(is_t) {
                    continue;
                }

                coord_t dx = cur.x - xs[i];
                coord_t disc = ls[i] * ls[i] - dx * dx;
                if(disc < 0) {
                    continue;
                }

                coord_t y_b = -sqrtl(disc);
                if(y_b < cur.y - 1e-12L && y_b > y_end) {
                    y_end = y_b;
                }
            }

            if(y_end < -1e17L) {
                y_end = target.y;
            }

            if(fabsl(cur.x - target.x) < 1e-9L && y_end < target.y - 1e-12L) {
                y_end = target.y;
            }

            Segment sg;
            sg.is_vert = true;
            sg.x = cur.x;
            sg.y0 = cur.y;
            sg.y1 = y_end;
            sg.len = cur.y - y_end;
            if(sg.len < 1e-12L) {
                break;
            }

            segs.push_back(sg);
            cur = Point(cur.x, y_end);
        } else {
            int arc_i = dr.arc_circle;
            coord_t theta0 = atan2l(cur.y, cur.x - xs[arc_i]);
            Point tang(-sinl(theta0), cosl(theta0));
            coord_t s = (tang.dot(dr.dir) > 0) ? 1.0L : -1.0L;
            coord_t best_dt = two_pi;
            Point best_pt = cur;
            bool have_event = false;
            for(int j: S) {
                if(j == arc_i) {
                    continue;
                }

                for(auto& q: circle_pair_inters(arc_i, j)) {
                    coord_t tq = atan2l(q.y, q.x - xs[arc_i]);
                    coord_t dt = fwd_angle(theta0, tq, s);
                    if(dt < 1e-9L) {
                        continue;
                    }
                    if(dt < best_dt) {
                        best_dt = dt;
                        best_pt = q;
                        have_event = true;
                    }
                }
            }

            {
                coord_t tq = -Point::PI / 2;
                coord_t dt = fwd_angle(theta0, tq, s);
                if(dt > 1e-9L && dt < best_dt) {
                    best_dt = dt;
                    best_pt = Point(xs[arc_i], -ls[arc_i]);
                    have_event = true;
                }
            }

            if(!have_event) {
                break;
            }

            coord_t theta1 = theta0 + s * best_dt;
            Segment sg;
            sg.is_vert = false;
            sg.cx = xs[arc_i];
            sg.r = ls[arc_i];
            sg.theta0 = theta0;
            sg.theta1 = theta1;
            sg.len = ls[arc_i] * best_dt;
            if(sg.len < 1e-12L) {
                break;
            }

            segs.push_back(sg);
            cur = best_pt;
        }

        if((cur - target).norm() < 1e-7L) {
            cur = target;
            break;
        }
    }

    return segs;
}

Point interpolate(const Phase& ph, coord_t s) {
    if(s <= 0) {
        return ph.start_p;
    }

    if(s >= ph.total_len - 1e-12L) {
        return ph.end_p;
    }

    coord_t acc = 0;
    for(const auto& sg: ph.segs) {
        if(s <= acc + sg.len + 1e-12L) {
            coord_t local = s - acc;
            coord_t frac = (sg.len > 1e-14L) ? local / sg.len : 0;
            if(sg.is_vert) {
                return Point(sg.x, sg.y0 + (sg.y1 - sg.y0) * frac);
            } else {
                coord_t theta = sg.theta0 + (sg.theta1 - sg.theta0) * frac;
                return Point(sg.cx + sg.r * cosl(theta), sg.r * sinl(theta));
            }
        }
        acc += sg.len;
    }

    return ph.end_p;
}

void read() {
    cin >> n >> m;
    xs.resize(n);
    ls.resize(n);
    for(auto& v: xs) {
        cin >> v;
    }

    for(auto& v: ls) {
        cin >> v;
    }

    cut_seq.resize(n - 1);
    for(auto& v: cut_seq) {
        cin >> v;
        --v;
    }

    ts.resize(m);
    for(auto& v: ts) {
        cin >> v;
    }
}

void solve() {
    // Each remaining rope i is the closed disk D_i = { p : (p.x - xs[i])^2
    // + p.y^2 <= ls[i]^2 } centered at (xs[i], 0). For any non-empty active
    // set S the stable position of the ball is the unique point of minimum
    // y inside the intersection of those disks. The KKT conditions of
    // minimising y over that intersection say at least one constraint is
    // active at the optimum, and at most two are needed in 2D; the
    // candidates are therefore the |S| circle bottoms (xs[i], -ls[i]) and
    // the at most two intersections of every pair of circles. We enumerate
    // all of them, keep those that lie inside every disk of S, and pick the
    // one with the smallest y.
    //
    // Cuts are processed in the given order. Cut 1 happens at t = 0; cut j
    // (j >= 2) happens at the instant the ball first becomes stationary
    // after cut j - 1. If a cut does not change the stable point it is
    // instantaneous.
    //
    // Between two consecutive equilibria the ball follows a perfectly
    // viscous gradient descent: its velocity is the projection of gravity
    // g = (0, -1) onto the feasible tangent cone at the current point.
    // Let T be the set of currently tight constraints (circles where
    // |p - c_i| = ls[i]); the tangent cone is { d : d . n_i <= 0 for i in T }
    // with outward normals n_i = (p - c_i) / ls[i]. The projection is
    // computed analytically:
    //   - if g satisfies all g . n_i <= 0, the velocity is g itself and the
    //     motion is straight down;
    //   - otherwise for each k in T we compute d_k = g - (g . n_k) n_k,
    //     which is the projection of g onto the hyperplane n_k = 0
    //     (i.e. the unit tangent to circle k at p that lowers y), and keep
    //     the one whose y-component is the smallest among those that still
    //     satisfy d . n_l <= 0 for every other l in T;
    //   - if no such d exists the cone has only the zero descent direction
    //     and the ball is at equilibrium.
    //
    // Each piece of the trajectory is therefore either a vertical segment
    // (interior motion until we touch the first circle from inside) or a
    // circular arc on some currently-tight rope. While on the arc of circle
    // i we follow it until the first angular event: either an intersection
    // with another active circle j or the bottom of circle i at theta =
    // -pi/2. Hitting an intersection turns it into a vertex and the next
    // iteration's projection chooses whether to transfer to the new circle,
    // re-enter the interior, or stop. The trajectory is built event by
    // event with at most O(n^2) events total.
    //
    // Each phase stores its starting time, total arc length, the precise
    // (start, end) equilibrium points, and the ordered list of vertical /
    // arc segments with their Euclidean lengths. Because the ball travels
    // at unit speed, the time spent in a phase equals its total length.
    // A query at time t locates the unique phase whose interval contains t,
    // walks the segment list by the local distance s = t - start_t,
    // interpolates linearly along the active piece (vertical or arc), and
    // returns the resulting point. After the final cut the ball stays at
    // the bottom of the last rope for all future times.

    vector<int> active(n);
    iota(active.begin(), active.end(), 0);

    Point cur = equilibrium(active);
    coord_t cur_time = 0;
    vector<Phase> phases;
    Point final_pos = cur;

    for(int ci = 0; ci < n - 1; ++ci) {
        int cut = cut_seq[ci];
        active.erase(find(active.begin(), active.end(), cut));

        Point new_eq = equilibrium(active);
        if((new_eq - cur).norm() < 1e-9L) {
            cur = new_eq;
            final_pos = cur;
            continue;
        }

        vector<Segment> segs = trace_path(cur, new_eq, active);
        coord_t total = 0;
        for(auto& sg: segs) {
            total += sg.len;
        }

        if(total < 1e-9L) {
            cur = new_eq;
            final_pos = cur;
            continue;
        }

        Phase ph;
        ph.start_t = cur_time;
        ph.total_len = total;
        ph.start_p = cur;
        ph.end_p = new_eq;
        ph.segs = std::move(segs);
        phases.push_back(std::move(ph));

        cur = new_eq;
        cur_time += total;
        final_pos = cur;
    }

    cout << fixed << setprecision(10);
    for(coord_t t: ts) {
        Point pos = final_pos;
        for(const auto& ph: phases) {
            if(t <= ph.start_t + ph.total_len + 1e-9L) {
                coord_t s = t - ph.start_t;
                if(s < 0) {
                    s = 0;
                }
                if(s > ph.total_len) {
                    s = ph.total_len;
                }
                pos = interpolate(ph, s);
                break;
            }
        }

        cout << pos.x << ' ' << pos.y << '\n';
    }
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    int T = 1;
    for(int test = 1; test <= T; test++) {
        read();
        solve();
    }

    return 0;
}
```

4. Python Solution

```python
import sys
import math
from dataclasses import dataclass

# Numerical tolerances.
EPS = 1e-12
PI = math.acos(-1.0)


@dataclass
class Point:
    # 2D point/vector.
    x: float = 0.0
    y: float = 0.0

    def __add__(self, other):
        # Vector addition.
        return Point(self.x + other.x, self.y + other.y)

    def __sub__(self, other):
        # Vector subtraction.
        return Point(self.x - other.x, self.y - other.y)

    def __mul__(self, c):
        # Scalar multiplication.
        return Point(self.x * c, self.y * c)

    def __truediv__(self, c):
        # Scalar division.
        return Point(self.x / c, self.y / c)

    def dot(self, other):
        # Dot product.
        return self.x * other.x + self.y * other.y

    def norm2(self):
        # Squared length.
        return self.x * self.x + self.y * self.y

    def norm(self):
        # Euclidean length.
        return math.sqrt(self.norm2())


@dataclass
class DirResult:
    # Chosen descent direction and circle to follow.
    dir: Point
    arc_circle: int


@dataclass
class Segment:
    # A segment is either vertical or a circular arc.
    is_vert: bool

    # Vertical segment data.
    x: float = 0.0
    y0: float = 0.0
    y1: float = 0.0

    # Arc data.
    cx: float = 0.0
    r: float = 0.0
    theta0: float = 0.0
    theta1: float = 0.0

    # Euclidean length.
    length: float = 0.0


@dataclass
class Phase:
    # One motion phase after a cut.
    start_t: float
    total_len: float
    start_p: Point
    end_p: Point
    segs: list


def inside_disks(p, S, xs, ls, tol=1e-7):
    # Check whether point p lies inside every disk indexed by S.
    for i in S:
        dx = p.x - xs[i]
        dy = p.y
        if dx * dx + dy * dy > ls[i] * ls[i] + tol:
            return False
    return True


def circle_pair_inters(i, j, xs, ls):
    # Compute intersection points of circles i and j.
    dx = xs[j] - xs[i]
    d = abs(dx)

    # Same center: no useful pair intersections for this solution.
    if d < 1e-14:
        return []

    r1 = ls[i]
    r2 = ls[j]

    # No intersection if circles are disjoint or one is inside the other.
    if d > r1 + r2 + 1e-12 or d < abs(r1 - r2) - 1e-12:
        return []

    # Distance from center i to chord midpoint.
    a = (r1 * r1 - r2 * r2 + d * d) / (2.0 * d)

    # Squared half-length of the chord.
    h2 = r1 * r1 - a * a

    # Avoid tiny negative values from floating-point errors.
    if h2 < 0:
        h2 = 0.0

    h = math.sqrt(h2)

    # Sign of direction from circle i to circle j.
    sgn = 1.0 if xs[j] > xs[i] else -1.0

    # x-coordinate of chord midpoint.
    mx = xs[i] + a * sgn

    # Two symmetric intersections.
    return [Point(mx, h), Point(mx, -h)]


def equilibrium(S, xs, ls):
    # Find the lowest point in the intersection of active disks.
    if not S:
        return Point(0.0, -1e18)

    candidates = []

    # Bottom point of every active circle.
    for i in S:
        candidates.append(Point(xs[i], -ls[i]))

    # Intersection points of every pair of active circles.
    for a in range(len(S)):
        for b in range(a + 1, len(S)):
            candidates.extend(circle_pair_inters(S[a], S[b], xs, ls))

    # Choose the valid candidate with minimum y.
    best = Point(0.0, 1e18)

    for c in candidates:
        if inside_disks(c, S, xs, ls) and c.y < best.y:
            best = c

    return best


def tight_set(p, S, xs, ls):
    # Return active ropes that are tight at point p.
    tight = []

    for i in S:
        dx = p.x - xs[i]
        dy = p.y
        d = math.sqrt(dx * dx + dy * dy)

        if abs(d - ls[i]) < 1e-6:
            tight.append(i)

    return tight


def descent_dir(p, tight, xs, ls):
    # Gravity direction.
    g = Point(0.0, -1.0)

    # No tight constraints means straight fall.
    if not tight:
        return DirResult(g, -1)

    normals = []

    # Compute outward normals of tight circles.
    for i in tight:
        r = Point(p.x - xs[i], p.y)
        rn = r.norm()

        if rn < 1e-14:
            normals.append(Point(0.0, -1.0))
        else:
            normals.append(r / rn)

    # Test whether gravity itself is feasible.
    g_ok = True
    for nk in normals:
        if g.dot(nk) > 1e-9:
            g_ok = False
            break

    if g_ok:
        tangent_circle = -1

        # If gravity is tangent to some tight circle, mark it as arc circle.
        for k, nk in enumerate(normals):
            if abs(g.dot(nk)) < 1e-9:
                tangent_circle = tight[k]
                break

        return DirResult(g, tangent_circle)

    # Otherwise try projections of gravity onto tangents of tight circles.
    best_dir = Point(0.0, 1.0)
    best_circle = -1
    found = False

    for k, nk in enumerate(normals):
        gn = g.dot(nk)

        # Projection g - (g dot n) n.
        d = Point(-gn * nk.x, -1.0 - gn * nk.y)

        dn = d.norm()

        if dn < 1e-14:
            continue

        d = d / dn

        # Must move downward.
        if d.y >= -1e-12:
            continue

        ok = True

        # Must be feasible for every other tight disk.
        for kk, nkk in enumerate(normals):
            if kk == k:
                continue

            if d.dot(nkk) > 1e-9:
                ok = False
                break

        if not ok:
            continue

        # Prefer the steepest downward direction.
        if not found or d.y < best_dir.y:
            best_dir = d
            best_circle = tight[k]
            found = True

    if not found:
        # No descent direction: already stable.
        return DirResult(Point(0.0, 0.0), -1)

    return DirResult(best_dir, best_circle)


def fwd_angle(frm, to, s):
    # Angular distance from frm to to in direction s = +1 or -1.
    two_pi = 2.0 * PI
    dt = s * (to - frm)
    dt = math.fmod(dt, two_pi)

    if dt < 0:
        dt += two_pi

    return dt


def trace_path(start, target, S, xs, ls):
    # Build vertical/arc segments from start to target.
    segs = []
    cur = start
    two_pi = 2.0 * PI

    # Large cap for safety.
    for _ in range(20000):
        if (cur - target).norm() < 1e-9:
            break

        tight = tight_set(cur, S, xs, ls)
        dr = descent_dir(cur, tight, xs, ls)

        if dr.dir.norm() < 1e-9:
            break

        if dr.arc_circle == -1:
            # Vertical free fall.
            y_end = -1e18

            for i in S:
                # Skip already tight ropes.
                if i in tight:
                    continue

                dx = cur.x - xs[i]
                disc = ls[i] * ls[i] - dx * dx

                if disc < 0:
                    continue

                y_b = -math.sqrt(disc)

                # First boundary encountered below current point.
                if y_b < cur.y - 1e-12 and y_b > y_end:
                    y_end = y_b

            # If no boundary is hit, go to target y.
            if y_end < -1e17:
                y_end = target.y

            # Clamp if target is on the same vertical line.
            if abs(cur.x - target.x) < 1e-9 and y_end < target.y - 1e-12:
                y_end = target.y

            length = cur.y - y_end

            if length < 1e-12:
                break

            segs.append(Segment(
                is_vert=True,
                x=cur.x,
                y0=cur.y,
                y1=y_end,
                length=length
            ))

            cur = Point(cur.x, y_end)

        else:
            # Move along an arc of circle arc_i.
            arc_i = dr.arc_circle

            theta0 = math.atan2(cur.y, cur.x - xs[arc_i])

            # Positive angular tangent.
            tang = Point(-math.sin(theta0), math.cos(theta0))

            # Choose angular direction matching descent direction.
            s = 1.0 if tang.dot(dr.dir) > 0 else -1.0

            best_dt = two_pi
            best_pt = cur
            have_event = False

            # Events: intersections with other active circles.
            for j in S:
                if j == arc_i:
                    continue

                for q in circle_pair_inters(arc_i, j, xs, ls):
                    tq = math.atan2(q.y, q.x - xs[arc_i])
                    dt = fwd_angle(theta0, tq, s)

                    if dt < 1e-9:
                        continue

                    if dt < best_dt:
                        best_dt = dt
                        best_pt = q
                        have_event = True

            # Event: bottom of this circle.
            tq = -PI / 2.0
            dt = fwd_angle(theta0, tq, s)

            if dt > 1e-9 and dt < best_dt:
                best_dt = dt
                best_pt = Point(xs[arc_i], -ls[arc_i])
                have_event = True

            if not have_event:
                break

            theta1 = theta0 + s * best_dt
            length = ls[arc_i] * best_dt

            if length < 1e-12:
                break

            segs.append(Segment(
                is_vert=False,
                cx=xs[arc_i],
                r=ls[arc_i],
                theta0=theta0,
                theta1=theta1,
                length=length
            ))

            cur = best_pt

        if (cur - target).norm() < 1e-7:
            cur = target
            break

    return segs


def interpolate(phase, s):
    # Return point after traveling distance s in this phase.
    if s <= 0:
        return phase.start_p

    if s >= phase.total_len - 1e-12:
        return phase.end_p

    acc = 0.0

    for sg in phase.segs:
        if s <= acc + sg.length + 1e-12:
            local = s - acc
            frac = local / sg.length if sg.length > 1e-14 else 0.0

            if sg.is_vert:
                # Linear interpolation on vertical segment.
                return Point(
                    sg.x,
                    sg.y0 + (sg.y1 - sg.y0) * frac
                )
            else:
                # Angular interpolation on circular arc.
                theta = sg.theta0 + (sg.theta1 - sg.theta0) * frac
                return Point(
                    sg.cx + sg.r * math.cos(theta),
                    sg.r * math.sin(theta)
                )

        acc += sg.length

    return phase.end_p


def solve():
    data = sys.stdin.read().strip().split()

    if not data:
        return

    ptr = 0

    n = int(data[ptr])
    ptr += 1

    m = int(data[ptr])
    ptr += 1

    xs = [float(data[ptr + i]) for i in range(n)]
    ptr += n

    ls = [float(data[ptr + i]) for i in range(n)]
    ptr += n

    # Convert cut order to zero-based indices.
    cut_seq = [int(data[ptr + i]) - 1 for i in range(n - 1)]
    ptr += n - 1

    ts = [float(data[ptr + i]) for i in range(m)]

    # Initially all ropes are active.
    active = list(range(n))

    # Initial equilibrium.
    cur = equilibrium(active, xs, ls)

    cur_time = 0.0
    phases = []
    final_pos = cur

    # Precompute all movement phases.
    for cut in cut_seq:
        # Remove cut rope.
        active.remove(cut)

        # New equilibrium after cutting.
        new_eq = equilibrium(active, xs, ls)

        # If unchanged, next cut is immediate.
        if (new_eq - cur).norm() < 1e-9:
            cur = new_eq
            final_pos = cur
            continue

        # Build trajectory.
        segs = trace_path(cur, new_eq, active, xs, ls)

        # Total duration equals path length.
        total = sum(sg.length for sg in segs)

        if total < 1e-9:
            cur = new_eq
            final_pos = cur
            continue

        phases.append(Phase(
            start_t=cur_time,
            total_len=total,
            start_p=cur,
            end_p=new_eq,
            segs=segs
        ))

        cur = new_eq
        cur_time += total
        final_pos = cur

    out = []

    # Answer queries.
    for t in ts:
        # If no phase contains t, ball is already at final position.
        pos = final_pos

        for ph in phases:
            if t <= ph.start_t + ph.total_len + 1e-9:
                s = t - ph.start_t

                if s < 0:
                    s = 0.0

                if s > ph.total_len:
                    s = ph.total_len

                pos = interpolate(ph, s)
                break

        out.append(f"{pos.x:.10f} {pos.y:.10f}")

    sys.stdout.write("\n".join(out))


if __name__ == "__main__":
    solve()
```

5. Compressed Editorial

Represent each rope as a disk centered at `(xi, 0)` with radius `li`. With active rope set `S`, the stable ball position is the lowest point in the intersection of these disks.

To find this point, enumerate:

- every circle bottom `(xi, -li)`,
- every pairwise circle intersection.

Keep candidates inside all active disks and choose minimum `y`.

After each cut, compute the new equilibrium. If it differs from the current point, trace the path.

At any point, tight ropes define tangent constraints `d . n_i <= 0` where `ni` is the outward unit normal. The ball wants gravity direction `(0, -1)` projected into this feasible tangent cone.

Thus motion is always either:

1. vertical downward, or
2. along an arc of a tight circle.

For vertical motion, continue until hitting the next disk boundary or the target equilibrium.

For arc motion, continue until hitting another circle intersection or the bottom of the current circle.

Store all segments of each phase. Since speed is `1`, segment length equals time. For each query, locate the phase and interpolate along its segments. If the time is after all phases, output the final equilibrium.
