
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 the general formula of the sequence $\{a_{n}\}$ as $a_{n}=2^{n-1}$, by keeping the order of the terms in the sequence $\{a_{n}\}$ unchanged, inserting $k(k\in N_{+})$ identical numbers $\left(-1\right)^{k}k$ between the $a_{k}$ and $a_{k+1}$ terms of the sequence $\{a_{n}\}$, forming a new sequence $\{b_{n}\}$. Let $T_{n}$ denote the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Find $T_{100}$.

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