
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Two players play the following game. From a pile containing 25 matches, each player on their turn takes one, two, or three matches. The winner is the one who ends up with an even number of matches after all matches have been taken.

a) Who wins with perfect play - the starting player or their partner? How should they play to win?

b) How does the answer change if the winner is the one who takes an odd number of matches?

c) Investigate this game in the general case, when there are $2n+1$ matches and any number of matches from 1 to $m$ can be taken.

Present the answer in LaTex format: \boxed{Your answer}
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