
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 $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, $\overrightarrow{m}=(\cos A+2\sin A,-3\sin A)$, $\overrightarrow{n}=(\sin A,\cos A-2\sin A)$,  
$(1)$ If $\overrightarrow{m} \parallel \overrightarrow{n}$ and angle $A$ is acute, find the size of angle $A$;  
$(2)$ Under the condition of $(1)$, if $\cos B= \frac {4}{5}$ and $c=7$, find the value of $a$.

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