
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 $C$: $\rho =2a\cos \theta (a > 0)$, line $l$: $\rho \cos \left( \theta -\frac{\pi }{3} \right)=\frac{3}{2}$, $C$ and $l$ have exactly one common point. $O$ is the pole, $A$ and $B$ are two points on $C$, and $\angle AOB=\frac{\pi }{3}$, then the maximum value of $|OA|+|OB|$ is __________.

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