
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
Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, and satisfy $S_{n+1}=S_{n}+a_{n}+2$ for $n\in \mathbb{N*}$, $2S_{5}=3(a_{4}+a_{6})$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If $b_{n}=a_{n}+(\frac{1}{2})^{a_{n}}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $T_{n}$.

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