
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
In the rectangular coordinate system \( xOy \), let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (\( a > 0, b > 0 \)), respectively. Point \( P \) is a point on the right branch of the hyperbola, \( M \) is the midpoint of \( PF_{2} \), and \( OM \perp PF_{2} \). Given that \( 3PF_{1} = 4PF_{2} \), find the eccentricity of the hyperbola.

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