
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
In the arithmetic sequence $\{a\_n\}$, $a\_2=6$. The sum of its first $n$ terms is $S\_n$. In the geometric sequence $\{b\_n\}$, all terms are positive, $b\_1=1$, and $b\_2+S\_4=33$, $b\_3=S\_2$.

1. Find $a\_n$ and $b\_n$.
2. Let the sum of the first $n$ terms of the sequence $\{c\_n\}$ be $T\_n$, where $c\_n=4b\_n-a\_5$. Find the smallest positive integer value of $n$ that makes the inequality $T\_n > S\_6$ true.

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