
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 the ellipse $\Gamma$ at points $P$ and $Q$. Given that $\left|P F_{2}\right|=\left|F_{1} F_{2}\right|$ and $3\left|P F_{1}\right|=4\left|Q F_{2}\right|$, determine the ratio of the minor axis to the major axis of the ellipse $\Gamma$.

[1] Note: This problem appeared in the 2014 National High School Mathematics Competition in China.

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