
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
Definition: If for any $n \in \mathbb{N}^*$, the sum of the first $n$ terms ($S_n$) of a sequence $\{ a_n \}$ is a perfect square, then the sequence $\{ a_n \}$ is called a "complete square sequence". Particularly, if there exists $n \in \mathbb{N}^*$ such that the sum of the first $n$ terms ($S_n$) of a sequence $\{ a_n \}$ is a perfect square, then the sequence $\{ a_n \}$ is called a "partial square sequence".

(1) If sequence $\{ a_n \}$ is a "partial square sequence", and $a_n = \begin{cases} 2, & n=1 \\ 2^{n-1}, & n \geqslant 2 \end{cases} \left( n \in \mathbb{N}^* \right)$, find the value of $n$ when the sum of the first $n$ terms ($S_n$) of the sequence $\{ a_n \}$ is a perfect square.

(2) If the sum of the first $n$ terms of sequence $\{ b_n \}$ is $T_n = (n-t)^2 \left( t \in \mathbb{N}^* \right)$, is the sequence $\{ | b_n | \}$ a "complete square sequence"? If yes, find the value of $t$; if not, explain the reason.

(3) Find all "complete square sequences" that are arithmetic progressions.

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