
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
In rectangle \( ABCD \), a circle \(\omega\) is constructed using side \( AB \) as its diameter. Let \( P \) be the second intersection point of segment \( AC \) with circle \(\omega\). The tangent to \(\omega\) at point \( P \) intersects segment \( BC \) at point \( K \) and passes through point \( D \). Find \( AD \), given that \( KD = 36 \).

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