
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
A point is chosen uniformly at random from the interior of a unit square. Let  $p$  be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that  $p^2$  can be written as  $m-\sqrt{n}$  for positive integers  $m$  and  $n$ , what is  $m+n$ ?


*2021 CCA Math Bonanza Lightning Round #3.1*

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