
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 standard equation of line $l$ is $x-y-2=0$, and the parametric equation of curve $C$ is 
$$
\begin{cases} 
x=2\sqrt{3}\cos \theta \\ 
y=2\sin \theta 
\end{cases}
(\theta \text{ is the parameter}).
$$
Suppose line $l$ intersects curve $C$ at points $A$ and $B$.

$(1)$ Find the length of segment $AB$.

$(2)$ Given that point $P$ moves on curve $C$. When the area of $\triangle PAB$ is maximized, find the coordinates of point $P$ and the maximum area of $\triangle PAB$.

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