
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 that the sequence $\{a_n\}$ is an arithmetic sequence with the first term $a_1=1$ and the common difference $d\neq0$. If $a_{b_1}, a_{b_2}, a_{b_3}, \ldots, a_{b_n}, \ldots$ form a geometric sequence, and $b_1=1, b_2=2, b_3=5$.

(1) Find the general formula for the sequence $\{b_n\}$.

(2) Let $c_n=\log_{3}(2b_n-1)$, calculate the sum $T_n=c_1c_2-c_2c_3+c_3c_4-c_4c_5+\ldots+c_{2n-1}c_{2n}-c_{2n}c_{2n+1}$.

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