
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, respectively. Let $M$ be a point on $C$ such that $MF_2$ is perpendicular to the x-axis, and the line $MF_1$ intersects $C$ at another point $N$.

$(1)$ If the slope of line $MN$ is $\frac{3}{4}$, find the eccentricity of $C$.

$(2)$ If the y-intercept of line $MN$ is $2$, and $|MN| = 5|F_1N|$, find $a$ and $b$.

$(3)$ Under the conditions of question $(2)$, let point $P$ move in the first quadrant of the ellipse. If the slope of line $F_2P$ is greater than $\frac{2}{\sqrt{3}}$, find the range of the slope of line $OP$ ($O$ is the origin).

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