
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 ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, whose left and right foci are $F_1(-\sqrt{2}, 0), F_2(\sqrt{2}, 0)$ respectively, and point $A(1, \frac{\sqrt{6}}{2})$ is on the ellipse $C$.

$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ A line passing through the fixed point $P(-1,0)$ intersects the ellipse $C$ at points $A$ and $B$. Given the fixed point $N(-\frac{7}{4},0)$, find the value of $\overrightarrow{NA} \cdot \overrightarrow{NB}$.

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