Let $p(n)$ be the number of ways to write $n$ as the sum of powers of two, ignoring order.
For example, $p(7) = 6$, the partitions being $$ \begin{align} 7 &= 1+1+1+1+1+1+1 \\ &=1+1+1+1+1+2 \\ &=1+1+1+2+2 \\ &=1+1+1+4 \\ &=1+2+2+2 \\ &=1+2+4 \end{align} $$ You are also given $p(7^7) \equiv 144548435 \pmod {10^9+7}$.
Find $p(7^{777})$. Give your answer modulo $10^9 + 7$.
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