
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
Label the vertices of a trapezoid  $T$  inscribed in the unit circle as  $A,B,C,D$  counterclockwise with  $AB\parallel CD$ . Let  $s_1,s_2,$  and  $d$  denote the lengths of  $AB$ ,  $CD$ , and  $OE$ , where  $E$  is the intersection of the diagonals of  $T$ , and  $O$  is the center of the circle. Determine the least upper bound of  $\frac{s_1-s_2}d$  over all  $T$  for which  $d\ne0$ , and describe all cases, if any, in which equality is attained.

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