
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 common foci of the ellipse $C_1: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)$ and the hyperbola $C_2: \frac{x^2}{a_1^2}-\frac{y^2}{b_1^2}=1 (a_1 > 0, b_1 > 0)$, they intersect at point $M$ in the first quadrant, with $\angle F_1MF_2=90^{\circ}$ and the eccentricity of the ellipse $C_1$ is $\frac{3}{4}$, find the eccentricity of the hyperbola $C_2$.

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