
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 $xOy$, the curve $C_{1}$ is defined by $\begin{cases} x=-2+\cos \alpha \\ y=-1+\sin \alpha \end{cases}$ (where $\alpha$ is a parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the curve $C_{2}$ is defined by $\rho\cos \theta-3=0$. Point $P$ is a moving point on the curve $C_{1}$.

$(1)$ Find the maximum distance from point $P$ to the curve $C_{2}$;

$(2)$ If the curve $C_{3}$: $\theta= \frac {\pi}{4}$ intersects curve $C_{1}$ at points $A$ and $B$, find the area of $\triangle ABC_{1}$.

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