
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 Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_{1}$ is $\rho\cos \theta=4$.

(Ⅰ) Point $M$ is a moving point on curve $C_{1}$, and point $P$ is on the line segment $OM$ and satisfies $|OM|\cdot|OP|=16$. Find the Cartesian equation of the trajectory $C_{2}$ of point $P$;

(Ⅱ) Suppose the polar coordinates of point $A$ are $\left(2,\frac{\pi}{3}\right)$, and point $B$ is on curve $C_{2}$. Find the maximum area of $\triangle OAB$.

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