
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 that the sequence $\{a\_n\}$ is an increasing sequence with the first term $0$, and the function $f\_n(x) = \left|\sin \frac{1}{n}(x - a\_n)\right|$, $x \in [a\_n, a\_{n - 1}]$ satisfies: for any real number $m \in [0, 1)$, $f\_n(x) = m$ always has two distinct roots. Find the general term formula for $\{a\_n\}$, i.e., $a\_n =$ _______ .

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