Balanceable $k$-bounded partitions (noch nicht ├╝bersetzt)

Problem 772

A $k$-bounded partition of a positive integer $N$ is a way of writing $N$ as a sum of positive integers not exceeding $k$.

A balanceable partition is a partition that can be further divided into two parts of equal sums.

For example, $3 + 2 + 2 + 2 + 2 + 1$ is a balanceable $3$-bounded partition of $12$ since $3 + 2 + 1 = 2 + 2 + 2$. Conversely, $3 + 3 + 3 + 1$ is a $3$-bounded partition of $10$ which is not balanceable.

Let $f(k)$ be the smallest positive integer $N$ all of whose $k$-bounded partitions are balanceable. For example, $f(3) = 12$ and $f(30) \equiv 179092994 \pmod {1\,000\,000\,007}$.

Find $f(10^8)$. Give your answer modulo $1\,000\,000\,007$.