
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
A curve $C$ in the rectangular coordinate system $xOy$ has the parametric equations $\begin{cases} x = a + \cos \beta \ y = a \sin \beta \end{cases}$ where $a > 0$ and $\beta$ is the parameter. Establish a polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the line $l$ is $\rho \cos \left( \theta - \frac{\pi}{3} \right) = \frac{3}{2}$.

(1) If the curve $C$ and line $l$ have only one common point, find the value of $a$.

(2) $A$ and $B$ are two points on the curve $C$ such that $\angle AOB = \frac{\pi}{3}$. Find the maximum area of $\triangle OAB$.

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