
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
Trapezoid \( EFGH \) has sides \( EF = 130 \), \( FG = 72 \), \( GH = 34 \), and \( EH = 100 \), with \( EF \) parallel to \( GH \). A circle with centre \( Q \) on \( EF \) is drawn tangent to \( FG \) and \( EH \). Given that \( EQ = \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers, find \( p+q \).

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