
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  $ABC$  be an acute triangle.  $PQRS$  is a rectangle with  $P$  on  $AB$ ,  $Q$  and  $R$  on  $BC$ , and  $S$  on  $AC$  such that  $PQRS$  has the largest area among all rectangles  $TUVW$  with  $T$  on  $AB$ ,  $U$  and  $V$  on  $BC$ , and  $W$  on  $AC$ . If  $D$  is the point on  $BC$  such that  $AD\perp BC$ , then  $PQ$  is the harmonic mean of  $\frac{AD}{DB}$  and  $\frac{AD}{DC}$ . What is  $BC$ ?

Note: The harmonic mean of two numbers  $a$  and  $b$  is the reciprocal of the arithmetic mean of the reciprocals of  $a$  and  $b$ .

*2017 CCA Math Bonanza Lightning Round #4.4*

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