
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 \( \triangle ABC \) be an acute scalene triangle with circumcircle \( \omega \). The tangents to \( \omega \) at \( B \) and \( C \) intersect at \( T \). Let \( X \) and \( Y \) be the projections of \( T \) onto lines \( AB \) and \( AC \), respectively. Suppose \( BT = CT = 20 \), \( BC = 30 \), and \( TX^2 + TY^2 + XY^2 = 2000 \). Find \( XY^2 \).

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