
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 the polar coordinate system, curve $C$: $\rho=4a\cos \theta(a > 0)$, line $l$: $\rho\cos \left( \theta- \frac{\pi}{3} \right)=4$, $C$ and $l$ have exactly one common point.

$(1)$ Find $a$;

$(2)$ Let $O$ be the pole, and $A$, $B$ be two points on curve $C$, with $\angle AOB= \frac{\pi}{3}$, find the maximum value of $|OA|+|OB|$.

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