
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 triangle $\triangle ABC$, $a=7$, $c=8$.  
$(Ⅰ)$ If $\overrightarrow{m}=(1,7a)$, $\overrightarrow{n}=(-4a,\sin C)$, and $\overrightarrow{m} \perp \overrightarrow{n}$, find the value of $\angle A$.  
$(Ⅱ)$ If the area of acute triangle $\triangle ABC$ is $16\sqrt{3}$, find the length of $b$.

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