
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 $P$ in the plane, the difference between the distance from point $P$ to point $F(1,0)$ and the distance from point $P$ to the $y$-axis is equal to $1$.

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

$(2)$ Draw two lines through point $F$ with existing slopes and perpendicular to each other, $l_{1}$ and $l_{2}$. Suppose $l_{1}$ intersects the trajectory $C$ at points $A$ and $B$, and $l_{2}$ intersects the 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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