
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
Given the parabola \( P: y^{2} = x \), the tangents at two moving points \( A \) and \( B \) on it intersect at point \( C \). Let \( D \) be the circumcenter of triangle \( \triangle ABC \). Aside from degenerate cases, does the circumcircle of triangle \( \triangle ABD \) pass through a fixed point? If so, find the fixed point, and if not, provide a counterexample.

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