
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
Within a square of side length $2$, $m$ small beads are randomly scattered (all falling within the square). Among them, $n$ beads fall within the inscribed circle of the square. A approximate value of $\pi$ can then be estimated as \_\_\_\_\_\_ (express in terms of $m$ and $n$).

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