
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
Given a sequence $\{a_n\}$, where $a_1 = \frac{1}{2}$, and for any positive integer $n$, $a_{n+1} = a_n + \frac{c}{a_n^2} > 0$. If $c = \frac{1}{216}$, determine whether there exists a positive integer $n$ such that $a_n > 2$. If it exists, find the smallest value of $n$; if not, explain why.

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