
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
In this problem assume \( s_{1} = 3 \) and \( s_{2} = 2 \). Determine, with proof, the nonnegative integer \( k \) with the following property:
1. For every board configuration with strictly fewer than \( k \) blank squares, the first player wins with probability strictly greater than \( \frac{1}{2} \); but
2. There exists a board configuration with exactly \( k \) blank squares for which the second player wins with probability strictly greater than \( \frac{1}{2} \).

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