
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 point $F(0,1)$, line $l$: $y=-1$, and $P$ as a moving point on the plane. Draw a line perpendicular to line $l$ passing through point $P$, with its foot $Q$. It is known that $\overrightarrow{QP} \cdot \overrightarrow{QF}=\overrightarrow{FP} \cdot \overrightarrow{FQ}$.

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

(2) Given that circle $M$ passes through the fixed point $D(0,2)$, and the center $M$ moves on the trajectory $C$. Circle $M$ intersects with the $x$-axis at points $A$ and $B$. Let $|DA|=l_{1}$ and $|DB|=l_{2}$. Find the maximum value of $\frac{l_{1}}{l_{2}}+\frac{l_{2}}{l_{1}}$.

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