
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
Let $\{a_{n}\}$ be a sequence of positive terms with the sum of the first $n$ terms denoted as $S_{n}$, $a_{2}=4$, and $4S_{n}=a_{n}^{2}+2a_{n}$.  
$(1)$ Find $a_{n}$;  
$(2)$ If ${b}_{n}=\frac{1}{S_{n}}$, find $\sum _{i=1}^{100}{b}_{i}$.

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