
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
The problem involves a curve $C_1$ in the rectangular coordinate system $xOy$ with the polar coordinate equation $\rho \cos \theta = 4$. Point $M$ moves on the curve $C_1$, and point $P$ is on the line segment $OM$ satisfying $|OM| \cdot |OP| = 16$. The tasks are:
(1) Find the rectangular coordinate equation of the trajectory $C_2$ of point $P$.
(2) Given point $A$ with polar coordinates $(2, \frac{\pi}{3})$, and point $B$ is on the curve $C_2$, find the maximum area of triangle $OAB$.

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