
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 $\{a\_n\}$ that satisfies: $a\_1=1$, $a\_{n+1}= \frac {1}{8} a\_n^2 + m$, where $n \in \mathbb{N}^*$, $m \in \mathbb{R}$.
(1) If $a\_1$, $m$, $a\_2$ form an arithmetic sequence, find the value of $m$;
(2) If $m=0$, find the general term $a\_n$ of the sequence $\{a\_n\}$;
(3) If for any positive integer $n$, $a\_n < 4$, find the maximum value of $m$.

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