
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
ABCD is a cyclic quadrilateral. The center of the circumcircle lies inside ABCD. The shortest side has length √(4 - t²) and the longest side has length t, where √2 < t < 2. The tangents at A and B intersect at A', the tangents at B and C intersect at B', the tangents at C and D intersect at C', and the tangents at D and A intersect at D'. Find the largest and smallest possible values for area A'B'C'D'/area ABCD.

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