
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 an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$, respectively. A line passing through $F_1$ and not perpendicular to either coordinate axis intersects $C$ at points $A$ and $B$, and the perimeter of $\triangle ABF_2$ is $8$. When the slope of line $AB$ is $\frac{3}{4}$, $AF_2$ is perpendicular to the $x$-axis.  
$(I)$ Find the equation of the ellipse $C$;  
$(II)$ Is there a fixed point $M$ on the $x$-axis such that $MF_1$ always bisects $\angle AMB$? Explain your reasoning.

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