
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 in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $4a = \sqrt{5}c$, $\cos C = \frac{3}{5}$.

$(Ⅰ)$ Find the value of $\sin A$.

$(Ⅱ)$ If $b = 11$,
$(i)$ Find the value of $a$.
$(ii)$ Find the value of $\cos \left(2A+C\right)$.

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