
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, $S\_n$, that satisfies $S\_n = n^2$. Another sequence $\{b\_n\}$ satisfies the following conditions: (1) $b\_3 = \frac{1}{4}$, (2) $b\_n > 0$, and (3) $b\_{n+1}^2 + b\_{n+1}b\_n - b\_n^2 = 0$.

1. Find the general terms for the sequences $\{a\_n\}$ and $\{b\_n\}$.
2. If $c\_n = a\_n b\_n$, find the sum of the first $n$ terms, $T\_n$, for the sequence $\{c\_n\}$.

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