
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 that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $a\sin A=b\sin B+(c-b)\sin C$.

(Ⅰ) Find $A$;

(Ⅱ) If the arithmetic sequence $\{a_n\}$ has a non-zero common difference, and $a_1\cos A=1$, 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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