
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 $$\begin{cases} x=1+2t \\ y= \frac {1}{2}-t\end{cases}$$ 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 two points $A$ and $B$.  
(1) Find $|AB|$;  
(2) Let $P$ be a point on curve $C$. When the area of $\triangle ABP$ is maximized, find the coordinates of point $P$.

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