
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, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established using the same unit of length. If the polar equation of curve $C_{1}$ is $\rho\sin\theta = -1$, and the parametric equation of curve $C_{2}$ is $\begin{cases} x = 2\cos\theta \\ y = -2 + 2\sin\theta \end{cases}$ (with $\theta$ as the parameter).

 $(1)$ Find the polar coordinates of the intersection points of $C_{1}$ and $C_{2}$;  
 $(2)$ Given the line $l: x - y + 2 = 0$, and point $P$ is on curve $C_{2}$, find the maximum distance from point $P$ to line $l$.

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