
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 $\Gamma$ with foci $F_{1}$ and $F_{2}$, and a line passing through point $F_{1}$ intersecting $\Gamma$ at points $P$ and $Q$. If $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|$, and $3\left|P F_{1}\right|=4\left|Q F_{1}\right|$, 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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