
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 the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with left and right foci $F_{1}$ and $F_{2}$ respectively, where the slope of the line $l$ intersecting the hyperbola $C$ at points $A$ and $B$ is $-3$, and point $M(4, -2\sqrt{2})$ lies on the hyperbola $C$ such that $|MF_{1}| \cdot |MF_{2}| = 24$.
- $(1)$ Find the area of $\triangle MF_{1}F_{2}$.
- $(2)$ If $\overrightarrow{OB} + \overrightarrow{OB'} = 0$ (where $O$ is the origin), and point $N(3,1)$, with the slopes of lines $NA$ and $NB'$ denoted as $k_{1}$ and $k_{2}$ respectively, is the product $k_{1} \cdot k_{2}$ a constant value? If so, determine this constant value. If not, explain why.

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