
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 two sequences: an arithmetic sequence $\{a_n\}$ and a geometric sequence $\{b_n\}$, where the sum of the first $n$ terms of the geometric sequence is $S_n$. The first term of the arithmetic sequence is $a_1 = -1$, the first term of the geometric sequence is $b_1 = 1$, the sum of the second terms of both sequences is 2, and the sum of the third terms of both sequences is 5.

(1) Find the general term formula and the sum of the first $n$ terms ($S_n$) for the geometric sequence $\{b_n\}$.

(2) Let $c_n = a_n \cdot \log_2 b_n$, find the sum of the first $n$ terms ($T_n$) for the sequence $\{c_n\}$. (Hint: $1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$)

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