
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
The sequence $\{a_n\}$ has a sum of its first $n$ terms denoted as $S_n$, with $a_1=t$ and $a_{n+1}=2S_n+1$ for all natural numbers $n$.
(1) For what value of $t$ is the sequence $\{a_n\}$ a geometric sequence?
(2) Under the condition of (1), if the sum of the first $n$ terms of the arithmetic sequence $\{b_n\}$, denoted as $T_n$, has a maximum value, with $T_3=15$, and $a_1+b_1$, $a_2+b_2$, $a_3+b_3$ form a geometric sequence, find $T_n$.

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