Low-Prime Chessboard Nim

Alice and Bob are taking turns playing a game consisting of $c$ different coins on a chessboard of size $n$ by $n$.

The game may start with any arrangement of $c$ coins in squares on the board. It is possible at any time for more than one coin to occupy the same square on the board at the same time. The coins are distinguishable, so swapping two coins gives a different arrangement if (and only if) they are on different squares.
On a given turn, the player must choose a coin and move it either left or up $2$, $3$, $5$, or $7$ spaces in a single direction. The only restriction is that the coin cannot move off the edge of the board.
The game ends when a player is unable to make a valid move, thereby granting the other player the victory.
Assuming that Alice goes first and that both players are playing optimally, let $M(n, c)$ be the number of possible starting arrangements for which Alice can ensure her victory, given a board of size $n$ by $n$ with $c$ distinct coins.
For example, $M(3, 1) = 4$, $M(3, 2) = 40$, and $M(9, 3) = 450304$.
What are the last $9$ digits of $M(10\,000\,019, 100)$?

This seems like it could be a question from a high-level competition such as the Olympiad or from a mathematical puzzle website such as Project Euler. However, those problems are usually creative and require inspiration to solve, which may not be achievable for AI as it designed to simulate human conversation, and not fully equipped to solve such high-level mathematical problems.

Nonetheless, some of the strategies that could be applied while attempting to solve such a problem could include:

1. Understanding how to classify the winning and losing states in terms of whose turn it is and the state of the board.

2. Succinctly representing the board state and the location of each coin.

3. Figuring out an efficient way to iterate over all possible states and transitions between states.

4. Implementing a dynamic programming algorithm to iteratively compute $M(n, c)$ for small values of $n$ and $c$ and then look for patterns or properties of the numbers obtained that could lead us to a way to efficiently compute $M(n, c)$ for larger values of $n$ and $c$.

Overall, this problem might involve ideas from combinatorics, game theory, dynamic programming, and modular arithmetic. However, it’s probably beyond the reach of most people (or AI) without a deep mathematical background. It is recommended to consult a math professor or an online math community for this specific and complex problem.

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