## 1. Abridged problem statement

There are `N` thimbles, and initially the ball is under thimble `1`. The croupier will perform exactly `M` given swap operations, each swapping two positions `Ai, Bi`, but the order of these `M` swaps can be chosen arbitrarily.

Determine all positions where the ball can end up after applying all swaps in some order.

Constraints: `2 ≤ N ≤ 100`, `1 ≤ M ≤ 1000`.

---

## 2. Detailed editorial

Model the swaps as an undirected multigraph:

- Vertices are thimble positions `1..N`.
- Each swap `(a, b)` is an undirected edge between `a` and `b`.
- Multiple swaps between the same pair become parallel edges.

When a swap edge is applied:

- If the ball is currently at one endpoint, the ball moves across that edge.
- Otherwise, the ball stays where it is.

So, for a chosen ordering of swaps, the edges that actually move the ball form an edge-disjoint walk, i.e. a **trail**, starting at vertex `1`. The final position is the endpoint of this trail.

The remaining edges are swaps that did not move the ball. Such an unused edge `{x, y}` can be inserted at some moment only if, at that moment, the ball is at a vertex different from both `x` and `y`.

Let `S(T)` be the set of vertices visited by a trail `T`.

An unused edge `{x, y}` can be ignored if and only if:

```text
S(T) \ {x, y} is non-empty
```

That is, the trail visits at least one vertex other than `x` and `y`.

Therefore:

- If a trail visits at least `3` distinct vertices, then every unused edge can be safely inserted somewhere, because an edge has only two endpoints.
- The hard cases are trails visiting only `1` or `2` vertices.

---

### Case 1: Target `u` is unreachable from `1`

If `u` is not connected to `1` in the graph, no trail can reach it.

So `u` is impossible.

---

### Case 2: `u != 1` and shortest distance from `1` to `u` is at least `2`

Then there is a path:

```text
1 -> ... -> u
```

with at least two edges, hence at least three distinct vertices.

This path itself is a valid trail visiting at least `3` vertices, so `u` is reachable.

---

### Case 3: `u != 1` and `u` is adjacent to `1`

Let `x` be the number of edges between `1` and `u`.

Temporarily remove all direct edges `{1, u}`.

#### Subcase 3.1: `1` can still reach `u`

Then there is an alternative path from `1` to `u`. Since direct edges were removed, this path has length at least `2`, so it visits at least `3` vertices.

Therefore `u` is reachable.

#### Subcase 3.2: `1` cannot reach `u` after removing `{1, u}`

Then any trail from `1` to `u` must cross between `1` and `u` using direct `{1, u}` edges.

If we use all `x` direct edges:

- If `x` is odd, we end at `u`.
- If `x` is even, we end back at `1`.

So if `x` is odd, `u` is reachable by oscillating between `1` and `u` using all direct edges.

If `x` is even, a simple oscillation is not enough. We need the trail to visit a third vertex. This is possible if either `1` or `u` belongs to some cycle after removing direct `{1, u}` edges.

The helper function `has_cycle_through(v)` checks whether there exists a cycle through vertex `v`. It does so by trying to remove one incident edge `{v, z}` and checking whether `v` can still reach `z`. If yes, then that removed edge plus the alternate path forms a cycle.

---

### Case 4: Target `u = 1`

We need a closed trail starting and ending at `1`.

There are three possibilities.

#### Possibility 1: Empty trail

If no edge is incident to vertex `1`, then all swaps avoid the ball, so it stays at `1`.

#### Possibility 2: Two-vertex oscillation

If there is a vertex `w` such that the multiplicity of edge `{1, w}` is even and at least `2`, then we can use all those edges to oscillate:

```text
1 -> w -> 1 -> w -> ... -> 1
```

This ends at `1`.

Unused edges can be inserted safely because the trail visits both `1` and `w`, and any unused edge different from `{1, w}` misses at least one of those two vertices.

#### Possibility 3: Closed trail visiting at least three vertices

A closed trail visiting at least three vertices exists if vertex `1` lies in a bridge-free component of size at least `3`.

Why bridges matter:

- A bridge cannot be part of an edge-disjoint closed trail, because crossing it would require crossing it back, reusing the same edge.
- Therefore closed trails live inside 2-edge-connected components.

The helper function `in_large_2ec(1)`:

1. Finds all bridges.
2. Removes them conceptually.
3. BFSes from `1` using only non-bridge edges.
4. Checks whether at least `3` vertices are reachable.

If yes, position `1` is possible.

---

### Complexity

`N ≤ 100`, so adjacency matrices are fine.

The implementation uses BFS several times over an `N × N` matrix.

Overall complexity is comfortably within limits, roughly polynomial around `O(N^4)` in the heaviest helper, but with `N = 100` this is fine.

---

## 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;
};

int n, m;
vector<vector<int>> g;

void read() {
    cin >> n >> m;
    g.assign(n + 1, vector<int>(n + 1, 0));
    for(int i = 0; i < m; i++) {
        int a, b;
        cin >> a >> b;
        g[a][b]++;
        g[b][a]++;
    }
}

bool reaches(int src, int dst) {
    vector<bool> vis(n + 1, false);
    queue<int> q;
    q.push(src);
    vis[src] = true;
    while(!q.empty()) {
        int v = q.front();
        q.pop();
        for(int w = 1; w <= n; w++) {
            if(g[v][w] > 0 && !vis[w]) {
                vis[w] = true;
                q.push(w);
            }
        }
    }

    return vis[dst];
}

vector<int> bfs_dist(int src) {
    vector<int> dist(n + 1, -1);
    queue<int> q;
    q.push(src);
    dist[src] = 0;
    while(!q.empty()) {
        int v = q.front();
        q.pop();
        for(int w = 1; w <= n; w++) {
            if(g[v][w] > 0 && dist[w] == -1) {
                dist[w] = dist[v] + 1;
                q.push(w);
            }
        }
    }

    return dist;
}

bool has_cycle_through(int v) {
    for(int z = 1; z <= n; z++) {
        if(z == v || g[v][z] == 0) {
            continue;
        }

        g[v][z]--;
        g[z][v]--;

        bool ok = reaches(v, z);

        g[v][z]++;
        g[z][v]++;

        if(ok) {
            return true;
        }
    }

    return false;
}

bool in_large_2ec(int v) {
    vector<vector<bool>> bridge(n + 1, vector<bool>(n + 1, false));
    for(int a = 1; a <= n; a++) {
        for(int b = a + 1; b <= n; b++) {
            if(g[a][b] != 1) {
                continue;
            }

            g[a][b] = g[b][a] = 0;
            bool reach = reaches(a, b);
            g[a][b] = g[b][a] = 1;

            if(!reach) {
                bridge[a][b] = bridge[b][a] = true;
            }
        }
    }

    vector<bool> vis(n + 1, false);
    queue<int> q;
    q.push(v);
    vis[v] = true;
    int cnt = 1;
    while(!q.empty()) {
        int u = q.front();
        q.pop();
        for(int w = 1; w <= n; w++) {
            if(g[u][w] > 0 && !bridge[u][w] && !vis[w]) {
                vis[w] = true;
                cnt++;
                q.push(w);
            }
        }
    }

    return cnt >= 3;
}

void solve() {
    // The M swaps form an undirected multigraph G on vertices 1...N. Whatever
    // ordering we pick, the ball traces a trail in G (an edge-disjoint walk):
    // a swap whose two positions include the ball moves it across that edge,
    // every other swap leaves the ball stationary. Conversely, given any
    // trail T from 1 to u with vertex set S(T), we can realise it as a full
    // ordering iff every edge {x,y} not in T can be slipped into some phase
    // where the ball sits at a vertex of S(T) outside {x,y}, i.e. iff
    // S(T) \ {x,y} is non-empty for every non-trail edge.
    //
    // This is automatic whenever |S(T)| >= 3, because no pair {x,y} can cover
    // three vertices. So position u is reachable as soon as we can produce a
    // trail from 1 to u that visits at least three distinct vertices, and the
    // hard cases are |S(T)| <= 2:
    //
    //   - dist[u] >= 2: the BFS shortest path is itself a trail with
    //     |S| >= 3, so u is reachable.
    //
    //   - dist[u] == 1 (a direct {1,u} edge exists): let x = mult({1,u}) and
    //     drop all {1,u} edges to form G'. If 1 still reaches u in G', the
    //     alternate path has length >= 2 and combined with a direct edge
    //     yields a |S| >= 3 trail. Otherwise the trail can only cross the
    //     {1,u} bridge through direct edges. Using all x of them oscillating
    //     ends at u iff x is odd, and that |S|=2 trail works because every
    //     other edge has at least one endpoint outside {1,u}. If x is even
    //     we must enrich the oscillation with a closed walk on a third
    //     vertex on either side, which exists iff 1 lies on a cycle in G'
    //     or u lies on a cycle in G' (multi-edges count, since we will glue
    //     a final direct {1,u} edge that already adds the third vertex).
    //
    //   - u == 1: we need a closed trail at 1 that absorbs every other edge.
    //     The empty trail handles degree(1) == 0; an oscillation on a
    //     multi-edge {1,w} of even multiplicity >= 2 handles |S| = 2 (every
    //     other edge has at least one endpoint outside {1,w}); and a closed
    //     trail visiting at least three distinct vertices handles |S| >= 3.
    //     A closed trail at 1 visiting >= 3 vertices exists iff 1 lies in a
    //     2-edge-connected component of size >= 3, which is *not* the same
    //     as 1 being on a simple cycle of >= 3 vertices: the trail may
    //     revisit a vertex (e.g. 1->2->3->2->1 in {1,2}*3 + {2,3}*2 reuses
    //     vertex 2 with edge-disjoint multi-edge pairs). We compute it by
    //     finding bridges -- single-multiplicity edges whose removal
    //     disconnects their endpoints -- and BFSing in the bridge-free
    //     subgraph, counting the vertices reachable from 1.
    //
    // Cycle-through-v in the dist[u] == 1 case allows multi-edges and is a
    // simple "remove one {v,z} edge, BFS, did we still reach z" check; it is
    // sufficient there because the final direct {1,u} edge already provides
    // the third vertex.

    vector<int> dist = bfs_dist(1);
    vector<int> ans;
    for(int u = 1; u <= n; u++) {
        if(dist[u] == -1) {
            continue;
        }

        if(u == 1) {
            int deg = 0;
            for(int w = 1; w <= n; w++) {
                deg += g[1][w];
            }

            if(deg == 0) {
                ans.push_back(u);
                continue;
            }

            bool even_multi = false;
            for(int w = 1; w <= n; w++) {
                if(g[1][w] >= 2 && g[1][w] % 2 == 0) {
                    even_multi = true;
                    break;
                }
            }

            if(even_multi || in_large_2ec(1)) {
                ans.push_back(u);
            }
        } else if(dist[u] >= 2) {
            ans.push_back(u);
        } else {
            int x = g[1][u];
            g[1][u] = g[u][1] = 0;
            bool ok = reaches(1, u);
            if(!ok && x % 2 == 1) {
                ok = true;
            }

            if(!ok && (has_cycle_through(1) || has_cycle_through(u))) {
                ok = true;
            }

            g[1][u] = g[u][1] = x;
            if(ok) {
                ans.push_back(u);
            }
        }
    }

    cout << ans << '\n';
}

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

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

    return 0;
}
```

---

## 4. Python solution with detailed comments

```python
from collections import deque
import sys


def main():
    # Read all integers from input.
    data = list(map(int, sys.stdin.read().split()))

    # First two integers are N and M.
    n, m = data[0], data[1]

    # Adjacency matrix with multiplicities.
    # g[a][b] = number of edges/swaps between a and b.
    g = [[0] * (n + 1) for _ in range(n + 1)]

    idx = 2

    # Read M swap operations.
    for _ in range(m):
        a = data[idx]
        b = data[idx + 1]
        idx += 2

        # Add undirected multiedge.
        g[a][b] += 1
        g[b][a] += 1

    def reaches(src, dst):
        """
        Return True if dst is reachable from src in the current graph.
        """
        visited = [False] * (n + 1)
        q = deque([src])
        visited[src] = True

        while q:
            v = q.popleft()

            # Scan all possible neighbors because we use adjacency matrix.
            for w in range(1, n + 1):
                if g[v][w] > 0 and not visited[w]:
                    visited[w] = True
                    q.append(w)

        return visited[dst]

    def bfs_dist(src):
        """
        Return shortest path distances from src.
        dist[v] == -1 means unreachable.
        """
        dist = [-1] * (n + 1)
        q = deque([src])
        dist[src] = 0

        while q:
            v = q.popleft()

            for w in range(1, n + 1):
                if g[v][w] > 0 and dist[w] == -1:
                    dist[w] = dist[v] + 1
                    q.append(w)

        return dist

    def has_cycle_through(v):
        """
        Check whether there is a cycle passing through vertex v.

        For every incident edge v-z, remove one copy.
        If v can still reach z, then that path plus the removed edge
        forms a cycle through v.
        """
        for z in range(1, n + 1):
            # Skip self and non-neighbors.
            if z == v or g[v][z] == 0:
                continue

            # Remove one copy of edge v-z.
            g[v][z] -= 1
            g[z][v] -= 1

            # If v still reaches z, the removed edge lies on a cycle.
            ok = reaches(v, z)

            # Restore the edge.
            g[v][z] += 1
            g[z][v] += 1

            if ok:
                return True

        return False

    def in_large_2ec(v):
        """
        Check whether v belongs to a bridge-free component of size >= 3.

        Such a component allows a closed trail through v visiting at least
        three distinct vertices.
        """
        bridge = [[False] * (n + 1) for _ in range(n + 1)]

        # Detect bridges naively.
        for a in range(1, n + 1):
            for b in range(a + 1, n + 1):
                # Only single edges can be bridges.
                # Parallel edges cannot be bridges.
                if g[a][b] != 1:
                    continue

                # Temporarily remove edge a-b.
                g[a][b] = 0
                g[b][a] = 0

                # If endpoints become disconnected, it is a bridge.
                if not reaches(a, b):
                    bridge[a][b] = True
                    bridge[b][a] = True

                # Restore edge.
                g[a][b] = 1
                g[b][a] = 1

        # BFS using only non-bridge edges.
        visited = [False] * (n + 1)
        q = deque([v])
        visited[v] = True
        count = 1

        while q:
            u = q.popleft()

            for w in range(1, n + 1):
                if g[u][w] > 0 and not bridge[u][w] and not visited[w]:
                    visited[w] = True
                    count += 1
                    q.append(w)

        return count >= 3

    # Distances from starting position 1.
    dist = bfs_dist(1)

    answer = []

    # Test every possible final position.
    for u in range(1, n + 1):
        # Unreachable vertices are impossible.
        if dist[u] == -1:
            continue

        # Special case: final position is 1.
        if u == 1:
            # Degree of vertex 1, counting multiplicities.
            deg = sum(g[1][w] for w in range(1, n + 1))

            # If no swap touches 1, the ball can remain at 1.
            if deg == 0:
                answer.append(u)
                continue

            # Check for even multiplicity edge {1, w}.
            even_multi = False
            for w in range(1, n + 1):
                if g[1][w] >= 2 and g[1][w] % 2 == 0:
                    even_multi = True
                    break

            # Either a two-vertex even oscillation works,
            # or a larger closed trail through 1 works.
            if even_multi or in_large_2ec(1):
                answer.append(u)

        # If u is at distance >= 2, a path to it visits at least 3 vertices.
        elif dist[u] >= 2:
            answer.append(u)

        # Remaining case: u is directly adjacent to 1.
        else:
            # Multiplicity of edge {1, u}.
            x = g[1][u]

            # Remove all direct edges between 1 and u.
            g[1][u] = 0
            g[u][1] = 0

            # Alternative path from 1 to u gives a 3-vertex trail.
            ok = reaches(1, u)

            # If no alternative path, using all direct edges ends at u
            # exactly when their count is odd.
            if not ok and x % 2 == 1:
                ok = True

            # If direct count is even, we need a cycle on either side
            # to visit a third vertex.
            if not ok and (has_cycle_through(1) or has_cycle_through(u)):
                ok = True

            # Restore direct edges.
            g[1][u] = x
            g[u][1] = x

            if ok:
                answer.append(u)

    # Output positions in any order.
    print(*answer)


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

---

## 5. Compressed editorial

Represent swaps as edges of an undirected multigraph. The ball moves exactly along the edges incident to its current vertex; these moving edges form a trail from `1` to the final vertex.

A trail `T` can be extended to an ordering of all swaps if every unused edge `{x,y}` can be performed while the ball is at some visited vertex not equal to `x` or `y`. If `T` visits at least `3` vertices, this is always true.

Therefore:

- If `u` is unreachable from `1`, impossible.
- If `u != 1` and `dist(1,u) >= 2`, possible.
- If `u` is adjacent to `1`:
  - Remove all `{1,u}` edges.
  - If `1` still reaches `u`, possible.
  - Else if multiplicity of `{1,u}` is odd, possible.
  - Else possible only if `1` or `u` lies on a cycle after removal.
- If `u = 1`:
  - Possible if `deg(1) = 0`.
  - Or if some edge `{1,w}` has even multiplicity at least `2`.
  - Or if `1` lies in a bridge-free component of size at least `3`.

Use BFS for reachability/distances, test cycles by removing one incident edge, and find bridge-free components by checking bridges.