
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 equation of line $l$ as $ \begin{cases} x=1+2t \\ y= \frac {1}{2}-t\\end{cases} $, and the parametric equation of curve $C$ as $ \begin{cases} x=2\cos θ \\ y=\sin θ\\end{cases} $, let points $A$ and $B$ be the intersection of line $l$ and curve $C$.

(1) Find $|AB|$;

(2) Let $P$ be a point on curve $C$. When the area of $\triangle ABP$ reaches its maximum value, find the coordinates of point $P$.

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