
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 an infinite sequence $\{a_n\}$, where $a_1=1$, $a_2=2$, and for any $n\in N^*$, $a_{n+2}=a_n$. The sequence $\{b_n\}$ satisfies $b_{n+1}-b_n=a_n$ ($n\in N^*$). If any term in the sequence $\left\{ \frac{b_{2n}}{a_n} \right\}$ repeats infinitely many times within the sequence, then the value of $b_1$ that meets the requirement is ______.

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