
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 that the distance from a moving point $P$ to the line $x=4$ is equal to twice the distance to the fixed point $F\_1(1,0)$,
(1) Find the trajectory equation of the moving point $P$;
(2) A line passing through $F\_1$ with a slope of $k=1$ intersects the above trajectory at points $C$ and $D$. If $A(2,0)$, find the area $S$ of $\triangle ACD$.

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