
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 $Ox$, the polar equation of curve $C$ is $p^{2}= \frac {144}{9+7\sin ^{2}\theta}$. Set up a Cartesian coordinate system with the pole $O$ as the origin and the polar axis as the positive $x$-axis.  
$(1)$ Find the Cartesian equation of the curve $C$;  
$(2)$ Suppose that the curve $C$ intersects the positive $x$-axis at point $A$ and the positive $y$-axis at point $B$. Let $P$ be a point on the curve $C$. Find the maximum value of the area of $\triangle ABP$.

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