
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 polar coordinates, the polar equation of curve $C_{1}$ is $\rho(\cos \theta + \sin \theta) = 4$. Now, with the pole $O$ as the origin and the positive x-axis as the polar axis, a plane Cartesian coordinate system is established. The parametric equation of curve $C_{2}$ is
$$
\begin{cases}
x = 2 + 3\cos \theta \\
y = 1 + 3\sin \theta
\end{cases}
\text{with } \theta \text{ as the parameter}.
$$
$(1)$ Find the rectangular coordinate equation of curve $C_{1}$ and the general equation of curve $C_{2}$;
$(2)$ If curves $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$, and $P$ is a moving point on curve $C_{2}$, find the maximum area of $\triangle PAB$.

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