
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 $\{a_n\}$ satisfy $a_n^2=a_{n+1}a_{n-1}+\lambda(a_2-a_1)^2$, where $n \geqslant 2$, $n \in \mathbb{N}$, and $\lambda$ is a constant.
1. If $\{a_n\}$ is an arithmetic sequence, and the common difference $d \neq 0$, find the value of $\lambda$;
2. If $a_1=1$, $a_2=2$, $a_3=4$, and there exists $r \in [3,7]$ such that $m \cdot a_n \geqslant n-r$ holds for any $n \in \mathbb{N}^*$, find the minimum value of $m$;
3. If $\lambda \neq 0$, and the sequence $\{a_n\}$ is not a constant sequence, and there exists a positive integer $T$, such that $a_{n+T}=a_n$ holds for any $n \in \mathbb{N}^*$. Find the minimum value of $T$ among all sequences $\{a_n\}$ that satisfy the condition.

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