
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
In a Cartesian coordinate system, points $F_1$ and $F_2$ are the left and right foci of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0, b > 0$), respectively. The eccentricity of the hyperbola $C$ is 2, and point $(1, \frac{3}{2})$ is on the hyperbola $C$. A moving point $P$ and another moving point $Q$ are symmetrical about the origin $O$ and are not on the $x$-axis. The perimeter of quadrilateral $PF_1QF_2$ is $4 \sqrt{2}$.

(1) Find the trajectory equation of the moving point $P$.
(2) Given a moving line $l$: $y = kx + m$ intersects the trajectory at two distinct points $M$ and $N$, and intersect the circle $W: x^2 + y^2 = \frac{3}{2}$ at two distinct points $G$ and $H$. As $m$ changes, the ratio $\frac{|MN|}{|GH|}$ remains constant. Find the value of the constant $k$.

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