
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  $ABCD$  be an isosceles trapezoid with  $AD\parallel BC$ .  Points  $P$  and  $Q$  are placed on segments  $\overline{CD}$  and  $\overline{DA}$  respectively such that  $AP\perp CD$  and  $BQ\perp DA$ , and point  $X$  is the intersection of these two altitudes.  Suppose that  $BX=3$  and  $XQ=1$ .  Compute the largest possible area of  $ABCD$ .

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