
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
Let the sequence $\{a_n\}$ be an increasing sequence with the first term being $0$, and let $f_n(x) = |\sin \frac{1}{n}(x-a_n)|$, where $x\in[a_n,a_{n+1}]$ and $n\in\mathbb{N}^*$, if for any $b\in[0,1)$, $f_n(x)=b$ always has two different real roots, determine the general formula for $\{a_n\}$.

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