1. Abridged Problem Statement

There are 10^6 cells in a row, initially empty. A tower is a maximal contiguous segment of non-empty cells. Towers are numbered from left to right; columns inside a tower are also numbered from left to right.

Process up to 10^6 commands:

- put x c: add c cubes to cell x.
- tput t x c: add c cubes to column x of tower t.
- towers: print number of towers.
- cubes t: print total cubes in tower t.
- length t: print length of tower t.
- tcubes t x: print cubes in column x of tower t.

All commands are valid. Output must exactly follow the sample format, including "1 towers", "1th", etc.

2. Detailed Editorial

Key observation

Cubes are only added, never removed.

Therefore:

- an empty cell can become non-empty;
- a non-empty cell stays non-empty forever;
- towers can appear, extend, or merge;
- towers never split.

A tower is just a maximal interval of positive-height cells.

For each tower we need to know:

- its starting cell;
- its ending cell;
- the total number of cubes inside it.

We also need to find the t-th tower from left to right quickly.

This suggests maintaining all current towers in an ordered balanced binary search tree keyed by starting position, augmented with subtree sizes. The submitted solution uses a randomized treap (here implemented via a generic, reusable Treap template).

Additionally, keep an array height[x] = number of cubes on cell x. Since cell indices are at most 10^6, this array is small enough.

Treap contents

Each treap node represents one tower.

For a tower [l, r], the node stores:
- key = l
- data.end = r
- data.sum = total cubes in cells l..r

The treap is ordered by key, i.e. by tower start position.

Each node also stores its subtree size, allowing us to find the k-th tower in O(log n) expected time.

Handling commands

towers
The number of towers is simply the size of the treap root: root ? root->size : 0.

length t
Find the t-th tower by order statistic. If its node is [l, r], length is r - l + 1.

cubes t
Find the t-th tower and output its stored sum.

tcubes t x
Find the t-th tower. Suppose it starts at l. Column x of this tower corresponds to global cell cell = l + x - 1. Then answer is height[cell].

tput t x c
Again map tower-local column x to a global cell. Then height[cell] += c and tower.sum += c. This operation never changes the tower structure because the command is guaranteed to refer to an existing column inside an existing tower.

put x c
This is the only command that can change tower structure. There are two cases.

Case 1: cell x is already non-empty
Then no interval boundaries change. We only add cubes to the cell and to the containing tower sum. The tower containing x is the tower with the greatest start position <= x, i.e. the predecessor by key:
  height[x] += c
  pred_node(root, x)->data.sum += c

Case 2: cell x is empty
Now it becomes non-empty. Check whether its immediate neighbors are non-empty:
  left = height[x - 1] > 0
  right = height[x + 1] > 0
There are four possibilities.

Neither neighbor is occupied
A new length-1 tower [x, x] appears: insert a new treap node with start=x, end=x, sum=c.

Only left neighbor is occupied
Cell x extends the tower ending at x - 1. Find the predecessor tower and set end = x, sum += c.

Only right neighbor is occupied
Cell x extends the tower starting at x + 1 to the left. Find the tower starting at x + 1; its new start becomes x. The solution directly changes the treap node key from x + 1 to x. This is safe because x - 1 is empty, so no previous tower touches x and no other tower start lies between x and x + 1, hence the BST order remains valid. Update key = x, sum += c.

Both neighbors are occupied
Cell x bridges two towers, which merge into one. Let the left tower be [l1, r1] with r1 = x - 1 and the right tower be [l2, r2] with l2 = x + 1. The new tower is [l1, r2]. Update the left tower node: end = right.end, sum += c + right.sum. Then erase the right tower node from the treap.

Complexity
Let T be the current number of towers. Each treap operation costs expected O(log T), so each command is processed in expected O(log T). Memory usage is O(10^6 + number_of_towers).

3. C++ Solution

```cpp
#include <bits/stdc++.h>
// #include <coding_library/data_structures/treap.hpp>

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;
};

template<class KeyT, class T, T (*merge_func)(T, T), uint64_t (*rng)()>
struct TreapNode {
    KeyT key;
    T data, subtree;
    uint64_t prior;
    size_t size;
    TreapNode *left, *right;

    TreapNode(KeyT key, T data)
        : key(key), data(data), left(nullptr), right(nullptr), size(1) {
        prior = rng();
    }

    void pull() {
        subtree = data;
        size = 1;
        if(left) {
            subtree = merge_func(left->subtree, subtree);
            size += left->size;
        }
        if(right) {
            subtree = merge_func(subtree, right->subtree);
            size += right->size;
        }
    }

    friend pair<TreapNode*, TreapNode*> split(TreapNode* t, KeyT key) {
        if(!t) {
            return {nullptr, nullptr};
        }
        if(key < t->key) {
            auto [left, t_left] = split(t->left, key);
            t->left = t_left;
            t->pull();
            return {left, t};
        } else {
            auto [t_right, right] = split(t->right, key);
            t->right = t_right;
            t->pull();
            return {t, right};
        }
    }

    friend pair<TreapNode*, TreapNode*> split_by_size(
        TreapNode* t, size_t size
    ) {
        if(!t) {
            return {nullptr, nullptr};
        }

        size_t left_size = t->left ? t->left->size : 0;
        if(left_size >= size) {
            auto [left, t_left] = split_by_size(t->left, size);
            t->left = t_left;
            t->pull();
            return {left, t};
        } else {
            auto [t_right, right] =
                split_by_size(t->right, size - 1 - left_size);
            t->right = t_right;
            t->pull();
            return {t, right};
        }
    }

    friend TreapNode* merge(TreapNode* l, TreapNode* r) {
        if(!l || !r) {
            return l ? l : r;
        } else if(l->prior > r->prior) {
            l->right = merge(l->right, r);
            l->pull();
            return l;
        } else {
            r->left = merge(l, r->left);
            r->pull();
            return r;
        }
    }

    friend TreapNode* unordered_merge(TreapNode* l, TreapNode* r) {
        if(!l) {
            return r;
        }
        if(!r) {
            return l;
        }
        if(l->prior < r->prior) {
            swap(l, r);
        }
        auto [t1, t2] = split(r, l->key);
        l->left = unordered_merge(l->left, t1);
        l->right = unordered_merge(l->right, t2);
        l->pull();
        return l;
    }

    friend void insert_in(TreapNode*& t, TreapNode* it) {
        if(!t) {
            t = it;
        } else if(it->prior > t->prior) {
            auto [t1, t2] = split(t, it->key);
            it->left = t1;
            it->right = t2;
            t = it;
        } else {
            insert_in(it->key < t->key ? t->left : t->right, it);
        }
        t->pull();
    }

    friend TreapNode* erase_from(
        TreapNode*& t, KeyT key, bool delete_node = false
    ) {
        T return_data;
        if(t->key == key) {
            auto tmp = t;
            t = merge(t->left, t->right);

            return_data = tmp->data;
            if(delete_node) {
                delete tmp;
                return_data = nullptr;
            }
        } else {
            return_data =
                erase_from(key < t->key ? t->left : t->right, key, delete_node);
        }
        if(t) {
            t->pull();
        }
        return return_data;
    }
};

template<class KeyT, class T, T (*merge_func)(T, T)>
class Treap {
  public:
    static uint64_t rng() {
        static mt19937_64 static_rng(random_device{}());
        return static_rng();
    }

    using Node = TreapNode<KeyT, T, merge_func, Treap::rng>;

    void _pull_all(Node* t) {
        if(t) {
            _pull_all(t->left);
            _pull_all(t->right);
            t->pull();
        }
    }

    Node* root;

    Treap() { root = nullptr; }
    Treap(const vector<pair<KeyT, T>>& a) { build_cartesian_tree(a); }

    void build_cartesian_tree(const vector<pair<KeyT, T>>& a) {
        vector<Node*> st;

        function<Node*(Node*)> recycle_stack = [&](Node* last) {
            Node* new_last = st.back();
            st.pop_back();
            new_last->right = last;
            return new_last;
        };

        for(const auto& [key, val]: a) {
            Node* new_node = new Node(key, val);
            Node* last = nullptr;
            while(!st.empty() && st.back()->prior < new_node->prior) {
                last = recycle_stack(last);
            }

            new_node->left = last;
            st.push_back(new_node);
        }

        root = nullptr;
        while(!st.empty()) {
            root = recycle_stack(root);
        }

        _pull_all(root);
    }

    void insert(KeyT key, T data) {
        Node* new_node = new Node(key, data);
        insert_in(root, new_node);
    }

    Node* erase(KeyT key) { return erase_from(root, key); }

    friend Treap<KeyT, T, merge_func> merge_treaps(
        Treap<KeyT, T, merge_func> l, Treap<KeyT, T, merge_func> r
    ) {
        Treap<KeyT, T, merge_func> res;
        res.root = unordered_merge(l.root, r.root);
        return res;
    }

    int count_leq(KeyT max_key) {
        int cnt = 0;
        Node* cur = root;
        while(cur) {
            if(cur->key <= max_key) {
                cnt += (cur->left ? cur->left->size : 0) + 1;
                cur = cur->right;
            } else {
                cur = cur->left;
            }
        }
        return cnt;
    }
};

struct Tower {
    int end;
    int64_t sum;
};

Tower merge_tower(Tower a, Tower b) { return a; }

using TowerTreap = Treap<int, Tower, merge_tower>;
using Node = TowerTreap::Node;

Node* kth_tower(Node* t, int k) {
    while(t) {
        int left_size = t->left ? (int)t->left->size : 0;
        if(k < left_size) {
            t = t->left;
        } else if(k == left_size) {
            return t;
        } else {
            k -= left_size + 1;
            t = t->right;
        }
    }
    return nullptr;
}

Node* find_node(Node* t, int key) {
    while(t) {
        if(t->key == key) {
            return t;
        }
        t = key < t->key ? t->left : t->right;
    }
    return nullptr;
}

Node* pred_node(Node* t, int x) {
    Node* res = nullptr;
    while(t) {
        if(t->key <= x) {
            res = t;
            t = t->right;
        } else {
            t = t->left;
        }
    }
    return res;
}

void erase_node(Node*& t, int key) {
    if(t->key == key) {
        Node* tmp = t;
        t = merge(t->left, t->right);
        delete tmp;
    } else {
        erase_node(key < t->key ? t->left : t->right, key);
    }
    if(t) {
        t->pull();
    }
}

namespace fastio {
constexpr int BUF_SIZE = 1 << 16;
char buf[BUF_SIZE];
int buf_pos = 0, buf_len = 0;

int get_char() {
    if(buf_pos == buf_len) {
        buf_len = (int)fread(buf, 1, BUF_SIZE, stdin);
        buf_pos = 0;
    }
    return buf_pos < buf_len ? (unsigned char)buf[buf_pos++] : EOF;
}

int skip_ws() {
    int c = get_char();
    while(c == ' ' || c == '\n' || c == '\r' || c == '\t') {
        c = get_char();
    }
    return c;
}

int read_int() {
    int c = skip_ws();
    int sign = 1;
    if(c == '-') {
        sign = -1;
        c = get_char();
    } else if(c == '+') {
        c = get_char();
    }
    int x = 0;
    while(c >= '0' && c <= '9') {
        x = x * 10 + (c - '0');
        c = get_char();
    }
    return sign * x;
}

int read_word(char* s) {
    int c = skip_ws();
    int len = 0;
    while(c != EOF && c != ' ' && c != '\n' && c != '\r' && c != '\t') {
        s[len++] = (char)c;
        c = get_char();
    }
    s[len] = 0;
    return len;
}

constexpr int OUT_SIZE = 1 << 22;
char out_buf[OUT_SIZE];
int out_pos = 0;

void flush_out() {
    fwrite(out_buf, 1, out_pos, stdout);
    out_pos = 0;
}

void put_char(char c) {
    if(out_pos == OUT_SIZE) {
        flush_out();
    }
    out_buf[out_pos++] = c;
}

void put_str(const char* s) {
    while(*s) {
        put_char(*s++);
    }
}

void put_int(int64_t x) {
    if(x < 0) {
        put_char('-');
        x = -x;
    }
    char tmp[24];
    int n = 0;
    if(x == 0) {
        tmp[n++] = '0';
    }
    while(x) {
        tmp[n++] = (char)('0' + x % 10);
        x /= 10;
    }
    while(n) {
        put_char(tmp[--n]);
    }
}

void put_ordinal(int64_t x) {
    put_int(x);
    put_str("th");
}
}  // namespace fastio

int n;
int height[1000002];
TowerTreap towers;

void do_put(int x, int cubes) {
    Node*& root = towers.root;
    if(height[x] > 0) {
        height[x] += cubes;
        pred_node(root, x)->data.sum += cubes;
        return;
    }

    height[x] = cubes;
    bool left = height[x - 1] > 0;
    bool right = height[x + 1] > 0;
    if(!left && !right) {
        towers.insert(x, Tower{x, (int64_t)cubes});
    } else if(left && !right) {
        Node* lt = pred_node(root, x - 1);
        lt->data.end = x;
        lt->data.sum += cubes;
    } else if(!left && right) {
        Node* rt = find_node(root, x + 1);
        rt->key = x;
        rt->data.sum += cubes;
    } else {
        Node* lt = pred_node(root, x - 1);
        Node* rt = find_node(root, x + 1);
        lt->data.end = rt->data.end;
        lt->data.sum += (int64_t)cubes + rt->data.sum;
        erase_node(root, x + 1);
    }
}

void read() { n = fastio::read_int(); }

void solve() {
    // Cubes are only ever added, never removed, so a filled cell stays filled
    // and a tower can only grow or merge with a neighbour - it never splits.
    // That lets us keep the towers in a treap keyed by their starting cell,
    // one node per tower, while a flat height[] array (cells go up to 10^6)
    // holds the cubes standing on each individual column.
    //
    // Each tower node stores its end cell and its total cube count, so 'cubes'
    // and 'length' of the t-th tower are answered by an order-statistic lookup
    // (the k-th tower by subtree size) reading a single node, and 'towers' is
    // just the number of nodes in the treap.
    //
    // A 'put x c' on an already filled cell only bumps that cell's height and
    // the sum of the tower containing it. On an empty cell we inspect the two
    // neighbours x-1 and x+1: with neither filled we insert a fresh length-one
    // tower; with only the left filled we extend that tower's end to x; with
    // only the right filled we relabel that tower's start down to x by editing
    // its node key, which keeps the BST order because x-1 is empty; with both
    // filled we bridge them, folding the right tower's end and sum into the
    // left node and erasing the right one. 'tput' and 'tcubes' map column x of
    // tower t to cell start+x-1 and reuse the per-cell height.

    char cmd[8];
    for(int i = 0; i < n; i++) {
        fastio::read_word(cmd);
        Node* root = towers.root;
        if(cmd[0] == 'p') {
            int x = fastio::read_int();
            int c = fastio::read_int();
            do_put(x, c);
        } else if(cmd[0] == 'c') {
            int t = fastio::read_int();
            Node* nd = kth_tower(root, t - 1);
            fastio::put_int(nd->data.sum);
            fastio::put_str(" cubes in ");
            fastio::put_ordinal(t);
            fastio::put_str(" tower\n");
        } else if(cmd[0] == 'l') {
            int t = fastio::read_int();
            Node* nd = kth_tower(root, t - 1);
            fastio::put_str("length of ");
            fastio::put_ordinal(t);
            fastio::put_str(" tower is ");
            fastio::put_int(nd->data.end - nd->key + 1);
            fastio::put_char('\n');
        } else if(cmd[1] == 'o') {
            fastio::put_int(root ? (int64_t)root->size : 0);
            fastio::put_str(" towers\n");
        } else if(cmd[1] == 'p') {
            int t = fastio::read_int();
            int x = fastio::read_int();
            int c = fastio::read_int();
            Node* nd = kth_tower(root, t - 1);
            height[nd->key + x - 1] += c;
            nd->data.sum += c;
        } else {
            int t = fastio::read_int();
            int x = fastio::read_int();
            Node* nd = kth_tower(root, t - 1);
            fastio::put_int(height[nd->key + x - 1]);
            fastio::put_str(" cubes in ");
            fastio::put_ordinal(x);
            fastio::put_str(" column of ");
            fastio::put_ordinal(t);
            fastio::put_str(" tower\n");
        }
    }

    fastio::flush_out();
}

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

    return 0;
}
```

4. Python Solution

```python
import sys
import random
from array import array

# ------------------------------------------------------------
# Fast scanner.
# It reads the whole input once and parses tokens manually.
# This avoids creating a huge list with split().
# ------------------------------------------------------------

class FastScanner:
    def __init__(self):
        self.data = sys.stdin.buffer.read()
        self.i = 0
        self.n = len(self.data)

    def next_token(self):
        data = self.data
        n = self.n
        i = self.i

        while i < n and data[i] <= 32:
            i += 1

        j = i
        while j < n and data[j] > 32:
            j += 1

        self.i = j
        return data[i:j]

    def next_int(self):
        data = self.data
        n = self.n
        i = self.i

        while i < n and data[i] <= 32:
            i += 1

        x = 0
        while i < n and data[i] > 32:
            x = x * 10 + data[i] - 48
            i += 1

        self.i = i
        return x


# ------------------------------------------------------------
# Treap stored in compact arrays.
#
# Node index 0 means null.
#
# For node v:
#   key[v]   = starting cell of tower
#   end[v]   = ending cell of tower
#   sumv[v]  = total cubes in tower
#   left[v]  = left child
#   right[v] = right child
#   size[v]  = subtree size
#   prio[v]  = random priority
# ------------------------------------------------------------

key = array("i", [0])
end = array("i", [0])
sumv = array("q", [0])
left = array("i", [0])
right = array("i", [0])
size = array("i", [0])
prio = array("Q", [0])

# Deterministic xorshift RNG for treap priorities.
seed = 88172645463393265


def rng():
    global seed
    seed ^= (seed << 7) & ((1 << 64) - 1)
    seed ^= seed >> 9
    return seed & ((1 << 64) - 1)


def new_node(k, r, s):
    """Create a new treap node and return its index."""
    idx = len(key)

    key.append(k)
    end.append(r)
    sumv.append(s)
    left.append(0)
    right.append(0)
    size.append(1)
    prio.append(rng())

    return idx


def pull(v):
    """Recalculate subtree size of node v."""
    size[v] = 1 + size[left[v]] + size[right[v]]


sys.setrecursionlimit(2_000_000)


def split(t, k):
    """
    Split treap t into:
      a: all nodes with key <= k
      b: all nodes with key > k
    """
    if t == 0:
        return 0, 0

    if k < key[t]:
        a, b = split(left[t], k)
        left[t] = b
        pull(t)
        return a, t
    else:
        a, b = split(right[t], k)
        right[t] = a
        pull(t)
        return t, b


def merge(a, b):
    """Merge two treaps where every key in a is <= every key in b."""
    if a == 0 or b == 0:
        return a or b

    if prio[a] > prio[b]:
        right[a] = merge(right[a], b)
        pull(a)
        return a
    else:
        left[b] = merge(a, left[b])
        pull(b)
        return b


def insert(t, v):
    """Insert node v into treap t and return new root."""
    if t == 0:
        return v

    if prio[v] > prio[t]:
        a, b = split(t, key[v])
        left[v] = a
        right[v] = b
        pull(v)
        return v

    if key[v] < key[t]:
        left[t] = insert(left[t], v)
    else:
        right[t] = insert(right[t], v)

    pull(t)
    return t


def erase(t, k):
    """Erase node with key k from treap t and return new root."""
    if key[t] == k:
        return merge(left[t], right[t])

    if k < key[t]:
        left[t] = erase(left[t], k)
    else:
        right[t] = erase(right[t], k)

    pull(t)
    return t


def kth(t, k):
    """Return index of zero-based k-th node by key order."""
    while t:
        ls = size[left[t]]

        if k < ls:
            t = left[t]
        elif k == ls:
            return t
        else:
            k -= ls + 1
            t = right[t]

    return 0


def find_node(t, k):
    """Find node with exact key k."""
    while t:
        if key[t] == k:
            return t
        if k < key[t]:
            t = left[t]
        else:
            t = right[t]
    return 0


def pred_node(t, x):
    """Find node with maximum key <= x."""
    ans = 0

    while t:
        if key[t] <= x:
            ans = t
            t = right[t]
        else:
            t = left[t]

    return ans


# ------------------------------------------------------------
# Main command processing.
# ------------------------------------------------------------

def main():
    scanner = FastScanner()

    n = scanner.next_int()

    # height[x] = cubes in global cell x.
    # Use 1_000_002 so x-1 and x+1 are safe for x in [1, 10^6].
    height = array("i", [0]) * 1_000_002

    root = 0

    out = []

    def ordinal(x):
        return str(x) + "th"

    def do_put(x, c):
        nonlocal root

        # If cell already belongs to a tower, only heights and sums change.
        if height[x] > 0:
            height[x] += c
            v = pred_node(root, x)
            sumv[v] += c
            return

        # Cell becomes non-empty.
        height[x] = c

        has_left = height[x - 1] > 0
        has_right = height[x + 1] > 0

        if not has_left and not has_right:
            # New isolated tower [x, x].
            root = insert(root, new_node(x, x, c))

        elif has_left and not has_right:
            # Extend the left tower to the right.
            lt = pred_node(root, x - 1)
            end[lt] = x
            sumv[lt] += c

        elif not has_left and has_right:
            # Extend the right tower to the left.
            rt = find_node(root, x + 1)
            key[rt] = x
            sumv[rt] += c

        else:
            # Merge left tower, new cell x, and right tower.
            lt = pred_node(root, x - 1)
            rt = find_node(root, x + 1)

            end[lt] = end[rt]
            sumv[lt] += c + sumv[rt]

            root = erase(root, x + 1)

    for _ in range(n):
        cmd = scanner.next_token()

        if cmd[0] == ord("p"):
            # put x c
            x = scanner.next_int()
            c = scanner.next_int()
            do_put(x, c)

        elif cmd[0] == ord("c"):
            # cubes t
            t = scanner.next_int()
            v = kth(root, t - 1)
            out.append(f"{sumv[v]} cubes in {ordinal(t)} tower\n")

        elif cmd[0] == ord("l"):
            # length t
            t = scanner.next_int()
            v = kth(root, t - 1)
            length = end[v] - key[v] + 1
            out.append(f"length of {ordinal(t)} tower is {length}\n")

        elif cmd[1] == ord("o"):
            # towers
            out.append(f"{size[root]} towers\n")

        elif cmd[1] == ord("p"):
            # tput t x c
            t = scanner.next_int()
            x = scanner.next_int()
            c = scanner.next_int()

            v = kth(root, t - 1)
            cell = key[v] + x - 1

            height[cell] += c
            sumv[v] += c

        else:
            # tcubes t x
            t = scanner.next_int()
            x = scanner.next_int()

            v = kth(root, t - 1)
            cell = key[v] + x - 1

            out.append(
                f"{height[cell]} cubes in {ordinal(x)} column of {ordinal(t)} tower\n"
            )

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


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

5. Compressed Editorial

Maintain every current tower as one node in an order-statistic treap keyed by its left endpoint. Each node stores start, end, total cubes, and subtree size. Also store per-cell cube counts in height[1..10^6].

Queries:
- towers: treap size.
- cubes t: find t-th node, print its sum.
- length t: find t-th node, print end - start + 1.
- tcubes t x: find t-th node, convert to global cell start + x - 1, print height[cell].

Updates:
- tput t x c: find t-th node, convert local column to global cell, add c to height[cell] and tower sum.
- put x c:
  - if height[x] > 0, add to cell and containing tower sum;
  - otherwise inspect height[x-1] and height[x+1]:
    - neither occupied: insert new tower [x, x];
    - only left occupied: extend left tower's end to x;
    - only right occupied: change right tower start from x+1 to x;
    - both occupied: merge left and right towers, erase right node.

Because cubes are never removed, towers never split, so these cases are sufficient.

Expected complexity per command: O(log number_of_towers). Memory: O(10^6 + number_of_towers).
