
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 that $\{a_{n}\}$ is an arithmetic sequence, and the sum of its first $n$ terms is $S_{n}$. If $a_{1}=2$ and $S_{7}=4(a_{2}+a_{5})$. Find:<br/>
$(1)$ The general formula for $\{a_{n}\}$;<br/>
$(2)$ Let ${b_n}=2{a_n}+{2^{{a_n}}}$. The sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$. Find $T_{n}$.

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