
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 increasing arithmetic sequence $\{a_{n}\}$, where $a_{1}=1$, $a_{2}$, $a_{4}+2$, $a_{14}$ form a geometric sequence.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ If $b_{n}=2^{a_{n}}$, find the sum of the first $n$ terms of $\{b_{n}\}$ denoted as $S_{n}$.

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