
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
Let the five consecutive positive integers be $n$, $n+1$, $n+2$, $n+3$, and $n+4$. Given that the sum of the second, third, and fourth integers is a perfect square and the sum of all five integers is a perfect cube, find the minimum possible value of $n+2$.

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