
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
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $\sqrt{3}a\cos C=(2b-\sqrt{3}c)\cos A$.

$(1)$ Find the magnitude of angle $A$;

$(2)$ Given an arithmetic sequence $\{a_n\}$ with a non-zero common difference, if $a_1\sin A=1$ and $a_2$, $a_4$, $a_8$ form a geometric sequence, find the sum of the first $n$ terms, $S_n$, of the sequence $\left\{ \frac{4}{a_na_{n+1}} \right\}$.

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