
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 $\{a_{n}\}$ and a geometric sequence $\{b_{n}\}$, where $a_{1}=1$ and ${a_{{b_n}}}={2^{n+1}}-1$. <br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$; <br/>$(2)$ Merge the terms in sequences $\{a_{n}\}$ and $\{b_{n}\}$, rearrange them in ascending order to form a new sequence $\{c_{n}\}$, and calculate the sum of the first $100$ terms of $\{c_{n}\}$.

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