
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
Chords \( \overline{AB} \) and \( \overline{CD} \) of circle \( \omega \) intersect at \( E \) such that \( AE = 8 \), \( BE = 2 \), \( CD = 10 \), and \( \angle AEC = 90^\circ \). Let \( R \) be a rectangle inside \( \omega \) with sides parallel to \( \overline{AB} \) and \( \overline{CD} \), such that no point in the interior of \( R \) lies on \( \overline{AB} \), \( \overline{CD} \), or the boundary of \( \omega \). What is the maximum possible area of \( R \)?

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