
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 with  $\angle ABC=60^{\circ}.$  Suppose points  $D$  and  $E$  are on lines  $AB$  and  $CB,$  respectively, such that  $CDB$  and  $AEB$  are equilateral triangles. Given that the positive difference between the perimeters of  $CDB$  and  $AEB$  is  $60$  and  $DE=45,$  what is the value of  $AB \cdot BC?$ *Proposed by Kyle Lee*

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