
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 ${a_n}$ with its first $n$ terms sum denoted as $S_n$, $a_1=t$, and the point $(S_n, a_{n+1})$ lies on the line $y=3x+1$, where $n \in \mathbb{N}^*$.
(1) For what real value of $t$ is the sequence ${a_n}$ a geometric sequence?
(2) Based on the conclusion of (1), let $b_n = \log_4 a_{2n}$, $c_n = b_n \times a_n$, and $T_n$ be the sum of the first $n$ terms of the sequence ${c_n}$. Find $T_n$.

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