
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 each term is positive, and $S_n$ is the sum of the first $n$ terms of this sequence, with $a_1=1$ and $2S_n=a_n \cdot a_{n+1}$ ($N \in \mathbb{N}^*$), find the minimum value of the positive integer $n$ that satisfies the inequality $\log_2(1+ \frac{1}{a_1})+\log_2(1+ \frac{1}{a_2})+\cdots+\log_2(1+ \frac{1}{a_n}) > 5$.

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