
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 all terms are positive, $a\_1=2$, and $a_{n+1} - a\_n = \frac{4}{a_{n+1} + a\_n}$, if the sum of the first $n$ terms of the sequence $\{\frac{1}{a_{n+1} + a\_n}\}$ is $5$, then $n=$________.

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