Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with Bob. The game begins with an arbitrary set of stone piles with a total number of stones equal to $n$.

During a player's turn, he/she must pick a pile having at least $2$ stones and perform a split operation, dividing the pile into an arbitrary set of $p$ non-empty, arbitrarily-sized piles where $2\le p\le k$ for some fixed constant $k$. For example, a pile of size $4$ can be split into $\{1,3\}$ or $\{2,2\}$, or $\{1,1,2\}$ if $k=3$ and in addition $\{1,1,1,1\}$ if $k=4$.

If no valid move is possible on a given turn, then the other player wins the game.

A winning position is defined as a set of stone piles where a player can ultimately ensure victory no matter what the other player does.

Let $f(n,k)$ be the number of winning positions for Alice on her first turn, given parameters $n$ and $k$. For example, $f(5,2)=3$ with winning positions $\{1,1,1,2\},\{1,4\},\{2,3\}$. In contrast, $f(5,3)=5$ with winning positions $\{1,1,1,2\},\{1,1,3\},\{1,4\},\{2,3\},\{5\}$.

Let $g(n)$ be the sum of $f(n,k)$ over all $2\le k\le n$. For example, $g(7)=66$ and $g(10)=291$.

Find $g(200)mod({10}^9+7)$.