
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 a sequence $\{a\_n\}$ that satisfies $2a_{n+1}=a_{n}+a_{n+2}$ for $n=1,2,3,...$, its first $n$ terms sum is denoted as $S_{n}$, and it is known that $a_{3}=5$ and $S_{6}=36$.

1. Find $a_{n}$;
2. It is known that the geometric sequence $\{b_{n}\}$ satisfies $b_{1}+b_{2}=1+a$ and $b_{4}+b_{5}=a^{3}+a^{4}$ ($a \neq -1$). Let $T_{n}$ be the first $n$ terms sum of the sequence $\{a_{n}\cdot b_{n}\}$. Find $T_{n}$.

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