Two positive integers $a$ and $b$ are $2$-friendly when $\gcd(a,b) = 2^t, t \gt 0$. For example, $24$ and $40$ are $2$-friendly because $\gcd(24,40) = 8 = 2^3$ while $24$ and $36$ are not because $\gcd(24,36) = 12 = 2^2\cdot 3$ is not a power of $2$.
Let $f(n)$ be the number of pairs, $(p,q)$, of positive integers with $1\le p\lt q\le n$ such that $p$ and $q$ are $2$-friendly. You are given $f(10^2) = 1031$ and $f(10^6) = 321418433$ modulo $1\,000\,000\,007$.
Find $f(10^{11})$ modulo $1\,000\,000\,007$.
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