(Analysis by Benjamin Qi)

First, discard all occurrences of $\times 1$ since they don't affect the answer. Also, if a program contains an occurrence of $\times 0$, discard the portion of the program before the last such occurrence.

Say that two instructions are of the same type if they are both $\times d$ or both $+s$. When do two interleaved programs produce the same expression? It turns out that this happens if and only if one interleaved program can be transformed into the other by repeatedly swapping two adjacent instructions of the same type, where one of the instructions belongs to Bessie and the other belongs to Elsie.

Therefore, the number of distinct expressions is precisely the number of interleaved programs that do not contain a pair of adjacent instructions of the same type where the first instruction belongs to Elsie and the second instruction belongs to Bessie, because every such program that contains such a pair can be uniquely transformed via a series of swaps into a program that does not contain such a pair (while there exists such a pair, swap the two instructions in the pair).

The full solution involves dynamic programming on a grid. In the code below, $\texttt{dp}[i][j][k]$ is the number of ways to interleave the first $i$ instructions of Bessie's program with the first $j$ instructions of Elsie's program such that the last instruction in the interleaving belongs to Bessie if $k=0$ or Elsie if $k=1$. $\texttt{dp}[i][j][k]$ is used to update both $\texttt{dp}[i][j+1]$ (if Elsie adds her $j$-th instruction to the end of the interleaving) and $\texttt{dp}[i+1][j]$ (if Bessie adds her $i$-th instruction to the end of the interleaving) unless Elsie last added to the interleaving and the $j-1$-th instruction of Elsie's program has the same type as the $i$-th instruction of Bessie's program.

The overall time complexity is proportional to the number of DP states, which is $O(N^2)$.

#include <bits/stdc++.h>
using namespace std;

template <class T> using V = vector<T>;

const int MOD = 1e9 + 7;
void mod_add(int &a, int b) { a = (a + b) % MOD; }

string _s;
cin >> _s;
for (char c : _s) {
if (c == '1') continue;
if (c == '0') s.clear();
if (c != '+') c = '2';
s += c;
}
}

int solve() {
int N;
cin >> N;
string A, B;
V<V<array<int, 2>>> dp((int)size(A) + 1,
V<array<int, 2>>((int)size(B) + 1));
int ans = 0;
auto upd = [&](int x, int y, int k, int v) {
if (x <= (int)size(A) && y <= (int)size(B))
};
dp.at(0).at(0).at(0) = 1;
for (int i = 0; i <= (int)size(A); ++i) {
for (int j = 0; j <= (int)size(B); ++j) {
for (int k : {0, 1}) {
int v = dp.at(i).at(j).at(k);
if (v == 0) continue;
if (i == (int)size(A) && j == (int)size(B)) mod_add(ans, v);
else {
upd(i, j + 1, 1, v);
if (k == 0) upd(i + 1, j, 0, v);
else {
assert(j > 0);
if (i < (int)size(A) && B.at(j - 1) != A.at(i))
upd(i + 1, j, 0, v);
}
}
}
}
}
return ans;
}

int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
int T;
cin >> T;
while (T--) cout << solve() << "\n";
}