
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
Given the parametric equations of line $L$ as $\begin{cases} & x=1+2t \\ & y=\dfrac{1}{2}-t \\ \end{cases}$, and the parametric equations of curve $C$ as $\begin{cases} & x=2\cos \theta \\ & y=\sin \theta \\ \end{cases}$, suppose line $L$ intersects curve $C$ at points $A$ and $B$,

$(1)$ Find $|AB|$;

$(2)$ Let $P$ be a point on curve $C$. When the area of $\Delta ABP$ is maximized, find the coordinates of point $P$.

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