
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, which intersects the positive half of the $y$-axis at point $P(0,b)$ and has its right focus at $F(c,0)$, with $O$ being the origin of coordinates, and $\tan \angle PFO = \frac{\sqrt{2}}{2}$.

$(1)$ Find the eccentricity $e$ of the ellipse $C$;  
$(2)$ Given points $M(1,0)$, $N(3,2)$, a line $l$ passing through point $M$ intersects the ellipse $C$ at points $C$ and $D$. Let the slopes of lines $CN$ and $DN$ be $k_1$ and $k_2$ respectively. If $k_1 + k_2 = 2$, find the equation of the ellipse $C$.

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