
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 $C$ is $\rho^{2}-2\rho\cos \theta-4\rho\sin \theta+4=0$, and the equation of line $l$ is $x-y-1=0$.  
$(1)$ Write the parametric equation of curve $C$;  
$(2)$ Find a point $P$ on curve $C$ such that the distance from point $P$ to line $l$ is maximized, and find this maximum value.

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