Balanceable $k$-bounded partitions

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 $f(30)\equiv 179092994(\bmod1000000007)$.

Find $f(10^8)$. Give your answer modulo $1000000007$.