
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 $\Gamma$ be an ellipse with two foci $F_{1}$ and $F_{2}$. A line passing through $F_{1}$ intersects $\Gamma$ at points $P$ and $Q$. Given that $|P F_{2}| = |F_{1} F_{2}|$, and $3|P F_{1}| = 4|Q F_{1}|$, find the ratio of the minor axis to the major axis of the ellipse $\Gamma$.

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