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Problem 772

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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 $f(30)\equiv 179092994(\mathrm{mod}1000000007)$.

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

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可平衡$k$-有界分拆

正整数$N$的分拆是指将$N$写作一系列正整数之和。若这些正整数均不超过$k$，则称之为$k$-有界分拆。若这些正整数能分成和相同的两组，则称之为可平衡分拆。

例如，$3+2+2+2+2+1$是$12$的一种可平衡$3$-有界分拆，因为$3+2+1=2+2+2$。反之，$3+3+3+1$是$10$的一种$3$-有界分拆，但不是可平衡分拆。

记$f(k)$是最小的正整数$N$，使得其所有的$k$-有界分拆都是可平衡分拆。例如，$f(3)=12$，$f(30)\equiv 179092994(\mathrm{mod}1000000007)$。

求$f({10}^8)$，并将你的答案对$1000000007$取余。

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