
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 an arithmetic sequence $\{a_{n}\}$ with a common difference of $d$, the sum of the first $n$ terms is $S_{n}$, where $a_{1}=1$, $d \gt 0$, and $a_{1}$, $a_{2}$, $S_{3}$ form a geometric sequence.  
$(Ⅰ)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(Ⅱ)$ Let ${b_n}={a_n}+{2^{{a_n}}}$. Find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted by $T_{n}$.

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