
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
For a sequence $\{a_n\}$, if there exists a positive integer $T$ such that for any positive integer $n$, $a_{n+T} = a_n$ holds, then the sequence $\{a_n\}$ is called a periodic sequence with period $T$. Let $b_1 = m$ ($0 < m < 1$), and for any positive integer $n$, we have
$$
b_{n+1} = \begin{cases} 
\frac{1}{b_n} & (0 < b_n \leq 1) \\
b_n - 1 & (b_n > 1)
\end{cases}
$$
If the sequence $\{b_n\}$ is a periodic sequence with period 5, then the value of $m$ can be $\boxed{\sqrt{2} - 1}$ (only one value of $m$ that satisfies the condition is required).

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