
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
In the sequence $\left\{a_{n}\right\}$, if $a_{n}^{2}-a_{n-1}^{2}=p$ (where $n \geq 2, n \in \mathbf{N^{*}}$, and $p$ is a constant), then $\left\{a_{n}\right\}$ is called an "equal variance sequence." Below are the judgments for the "equal variance sequence":

1. The sequence $\left\{(-1)^{n}\right\}$ is an equal variance sequence;
2. If $\left\{a_{n}\right\}$ is an equal variance sequence, then $ \left\{a_{n}^{2}\right\}$ is an arithmetic sequence;
3. If $\left\{a_{n}\right\}$ is an equal variance sequence, then $\left\{a_{k n}\right\}\left(k \in \mathbf{N}^{*}, k\right.$ being a constant) is also an equal variance sequence;
4. If $\left\{a_{n}\right\} $ is both an equal variance sequence and an arithmetic sequence, then this sequence is a constant sequence.

Among them, the correct proposition numbers are $\qquad$ (Fill in all the correct proposition numbers on the line).

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