
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
Each of  $n$  boys and  $n$  girls chooses a random number from the set  $\{ 1, 2, 3, 4, 5 \}$ , uniformly and independently.  Let  $p_n$  be the probability that every boy chooses a different number than every girl.  As  $n$  approaches infinity, what value does  $\sqrt[n]{p_n}$  approach?

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