
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
In the diagram, \(ABCD\) is a trapezoid with \(\angle DAB = \angle ADC = 90^\circ\). Also, \(AB = 7\), \(DC = 17\), and \(AD = 10\). Point \(P\) is on \(AD\) and point \(Q\) is on \(BC\) so that \(PQ\) is parallel to \(AB\) and to \(DC\). Point \(F\) is on \(DC\) so that \(BF\) is perpendicular to \(DC\). \(BF\) intersects \(PQ\) at \(T\).

(a) Determine the area of trapezoid \(ABCD\).

(b) Determine the measure of \(\angle BQP\).

(c) If \(PQ = x\), determine the length of \(AP\) in terms of \(x\).

(d) Determine the length of \(PQ\) for which the areas of trapezoids \(ABQP\) and \(PQCD\) are equal.

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