
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 the Cartesian coordinate system $\alpha$, point $A$ is the right vertex of the ellipse $C: \frac{x^2}{4} + \frac{y^2}{b^2} = 1 (0 < b < 2)$. A line $l$ passing through $O$ with a non-zero slope intersects $C$ at points $P$ and $Q$. The lines $AP$ and $AQ$ intersect the axis $\left( -\frac{1}{2}, -\frac{1}{3} \right) \cup \left( \frac{1}{2}, 1 \right)$ at points $M$ and $N$, respectively.

$(1)$ When $t_1 = 0, t_2 = \frac{4\sqrt{2}}{3}$, if $|PQ| = 2|AP|$, find the equation of $C$;

$(2)$ If $b$ is a constant, determine whether $|OM| \cdot |ON|$ is a constant value? And explain the reason.

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