
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
From the set $M = \{(x,y) \mid (|x|-1)^{2} + (|y|-1)^{2} < 4\}$, where $x$ and $y$ are integers, randomly select a point $P(x,y)$. If the probability that $xy \geqslant k$ (where $k > 0$) is $\frac{6}{25}$, then the maximum value of $k$ is ______.

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