
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 an arithmetic sequence $\{a_{n}\}$ where all terms are distinct, $a_{1}=1$, and $a_{1}$, $a_{2}$, $a_{5}$ form a geometric sequence. The sum of the first $n$ terms of sequence $\{b_{n}\}$ is $S_{n}=2^{n+1}-2$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Let ${c_n}={2^{{a_n}}}+{log_2}{b_n}$, find the sum of the first $n$ terms of sequence $\{c_{n}\}$ denoted as $T_{n}$.

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