
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 $F(0, 1)$, a line $l: y = -1$, and a moving point $P$ on the plane such that a perpendicular line from $P$ to line $l$ meets at point $Q$, and $\overrightarrow{QP} \cdot \overrightarrow{QF} = \overrightarrow{FP} \cdot \overrightarrow{FQ}$.
(1) Find the equation for the locus of point $P$, denoted as $C$.
(2) A circle $M$ passes through a fixed point $D(0, 2)$ with its center $M$ moving on the locus $C$, and it intersects 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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