
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 a sequence $\{a_{n}\}$ where all terms are positive numbers and satisfy $a_{1}=1$, $a_{n+1}^{2}-2S_{n}=n+1$ where $n\in \mathbb{N}^{*}$, and $S_{n}$ is the sum of the first $n$ terms of the sequence.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Another sequence $\{b_{n}\}$ is defined as $b_{n}=a_{n} \cdot \sin\left(\frac{n\pi}{2}\)$. Find the sum of the first $100$ terms of $\{b_{n}\}$ denoted as $T_{100}$.

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