Let $s_k$ be the number of 1’s when writing the numbers from 0 to $k$ in binary.
For example, writing 0 to 5 in binary, we have $0,1,10,11,100,101$. There are seven 1’s, so $s_5=7$.
The sequence $S=\{s_k:k\ge 0\}$ starts $\{0,1,2,4,5,7,9,12,...\}$.

A game is played by two players. Before the game starts, a number $n$ is chosen. A counter $c$ starts at 0. At each turn, the player chooses a number from 1 to $n$ (inclusive) and increases $c$ by that number. The resulting value of $c$ must be a member of $S$. If there are no more valid moves, then the player loses.

For example, with $n=5$ and starting with $c=0$:

Player 1 chooses 4, so $c$ becomes $0+4=4$.
Player 2 chooses 5, so $c$ becomes $4+5=9$.
Player 1 chooses 3, so $c$ becomes $9+3=12$.
etc.

Note that $c$ must always belong to $S$, and each player can increase $c$ by at most $n$.

Let $M(n)$ be the highest number that the first player could choose at the start to force a win, and $M(n)=0$ if there is no such move. For example, $M(2)=2$, $M(7)=1$, and $M(20)=4$.

It can be verified that $\sum M(n{)}^3=8150$ for $1\le n\le 20$.

Find $\sum M(n{)}^3$ for $1\le n\le 1000$.