
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 the terms of the sequence $\{a_{n}\}$ are all positive numbers, and $a_{1}=1$, for any $n\in N^{*}$, $a_{n+1}=2a_{n}+1$, $b_{n}=2\log _{2}(1+a_{n})-1$. If the terms of sequence $\{a_{n}\}$ are removed from the sequence $\{b_{n}\}$, the remaining terms form a sequence $\{c_{n}\}$. Determine the sum of the first one hundred terms of the sequence $\{c_{n}\}$.

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