
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
The sequence \(\{a_n\}\) is defined as follows:
\[ a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{[a_n]}, \, n \geq 1, \]
where \([a_n]\) is the integer part of \(a_n\). For which values of \(n\) does the inequality \(a_n > 20\) necessarily hold?

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