
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 $a_1=1$, and the sum of the first $n$ terms is $S_n$, with $S_n= \frac{1}{2}a_na_{n+1}$. If the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{2S_n} \right\}$ is $T_n= \frac{99}{100}$, then $n=$ ______.

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