
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 that $F_1$ and $F_2$ are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $P\left( 1, \frac{3}{2} \right)$ lies on it, with $PF_1 + PF_2 = 4$.

$(1)$ Find the standard equation of the ellipse $C$.

$(2)$ Let circle $O$ have $F_1$ and $F_2$ as its diameter. Line $l: y = kx + m$ is tangent to circle $O$ and intersects ellipse $C$ at two distinct points $A$ and $B$. If $\overrightarrow{OA} \cdot \overrightarrow{OB} = -\frac{3}{2}$, find the value of $k$.

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