
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 sequences $\{a_{n}\}$ and $\{b_{n}\}$, where $a_{1}=100$, $a_{n+1}=a_n^2$, and $b_{n}=\lg a_{n}$.  
$(1)$ Find the general formula for sequence $\{b_{n}\}$.  
$(2)$ If $c_{n}=\log _{2}b_{n}+\log _{2}b_{n+1}+\cdots +\log _{2}b_{2n}$, find the sum $S_{n}$ of the first $n$ terms of the sequence $\left\{\frac{1}{c_n}\right\}$.

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