
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 $a_n^2 - a_{n+1}a_{n-1} = m(a_2 - a_1)^2$, where $n \geq 2$, $n \in \mathbb{N}^*$, and $m$ is a constant.

(1) If $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, find the value of $m$.

(2) If $a_1 = 1$, $a_2 = 2$, $a_3 = 4$, and there exists $p \in [3,5]$ such that $t \cdot a_n + p \geq n$ holds for any $n \in \mathbb{N}^*$, find the minimum value of $t$.

(3) If $m \neq 0$, and $\{a_n\}$ is not a constant sequence. If there exists a positive integer $T$ such that $a_{n+T} = a_n$ holds for any $n \in \mathbb{N}^*$, find the smallest value of $T$ that satisfies the conditions.

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