Two positive integers $a$ and $b$ are $2$-friendly when $gcd(a,b)=2^t,t>0$. 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 $1\le p<q\le n$ such that $p$ and $q$ are $2$-friendly. You are given $f({10}^2)=1031$ and $f({10}^6)=321418433$ modulo $1000000007$.

Find $f({10}^{11})$ modulo $1000000007$.