
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 a moving point $P$ on a plane, the distance from point $P$ to point $F(1,0)$ is equal to the distance from point $P$ to the line $x=-1$.

(1) Find the equation of the trajectory $C$ of the moving point $P$;

(2) Draw two lines $l\_1$ and $l\_2$ passing through point $F$ with existing slopes and perpendicular to each other. Suppose $l\_1$ intersects with trajectory $C$ at points $A$ and $B$, and $l\_2$ intersects with trajectory $C$ at points $D$ and $E$. Find the minimum value of $\overrightarrow{AD} \cdot \overrightarrow{EB}$.

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