
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
Two chords, $EF$ and $GH,$ intersect inside a circle at $Q.$ If $EQ = 5$ and $GQ = 7,$ find the length of segments $FQ$ and $HQ$ and then calculate the ratio \( \frac{FQ}{HQ} \).

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