Gyrating Glyphs

문제

You are rocking the latest breakthrough in Computer Science: animated fonts. Suddenly, all of your colleagues' code looks amazing, and you are finally motivated to review it. Unfortunately, due to the constant rotations, it is hard to distinguish between the $++$ (plus) and the $\times \backslash times$ (multiply) operators (all the other characters are still readable). The function you are reviewing takes as input $n+1n+1$ integers $a_0,a_1,\dots ,a_{n}a\_0,\; a\_1,\; \backslash ldots,\; a\_n$ and returns the value $$(\dots (((a_0{\mathrm{op}}_1{a}_1){\mathrm{op}}_2{a}_2){\mathrm{op}}_3{a}_3)\dots {\mathrm{op}}_n{a}_n){10}^9+7,\backslash bigg(\backslash ldots\backslash Big(\backslash big((a\_0\; \backslash ,\backslash operatorname\{op\}\_1\backslash ,\; a\_1)\; \backslash ,\backslash operatorname\{op\}\_2\backslash ,\; a\_2\backslash big)\; \backslash ,\backslash operatorname\{op\}\_3\backslash ,\; a\_3\backslash Big)\; \backslash ldots\; \backslash ,\backslash operatorname\{op\}\_n\backslash ,\; a\_n\backslash bigg)\backslash quad\; \backslash bmod\; 10^9+7,$$ where the $nn$ operators ${\mathrm{op}}_1,{\mathrm{op}}_2,\dots ,{\mathrm{op}}_n\backslash operatorname\{op\}\_1,\backslash ,\; \backslash operatorname\{op\}\_2,\backslash ,\; \backslash ldots,\backslash ,\; \backslash operatorname\{op\}\_n$ are either $++$ or $\times \backslash times$. For example when given input $(a_0,a_1,a_2)=(1,1,2)(a\_0,a\_1,a\_2)\; =\; (1,1,2)$ with hidden operators $({\mathrm{op}}_1,{\mathrm{op}}_2)=(+,\times )(\backslash operatorname\{op\}\_1,\backslash operatorname\{op\}\_2)=(+,\backslash times)$, then the function returns $((1+1)\times 2)=4{10}^9+7((1+1)\backslash times2)=4\; \backslash bmod\; 10^9+7$.

You can still execute the function a few times on some input and read the returned value. Use this to recover the operators.