
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
Given  $\triangle{ABC}$  with  $\angle{B}=60^{\circ}$  and  $\angle{C}=30^{\circ}$ , let  $P,Q,R$  be points on the sides  $BA,AC,CB$  respectively such that  $BPQR$  is an isosceles trapezium with  $PQ \parallel BR$  and  $BP=QR$ .
Find the maximum possible value of  $\frac{2[ABC]}{[BPQR]}$  where  $[S]$  denotes the area of any polygon  $S$ .

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