
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
Mike and Harry play a game on an \(8 \times 8\) board. For some positive integer \(k\), Mike chooses \(k\) squares and writes an \(M\) in each of them. Harry then chooses \(k+1\) squares and writes an \(H\) in each of them. After Harry is done, Mike wins if there is a sequence of letters forming "HMM" or "MMH," when read either horizontally or vertically, and Harry wins otherwise. Determine the smallest value of \(k\) for which Mike has a winning strategy.

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