
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 a point \( M(1,1) \) on the parabola \( y = x^{2} \) as the right-angle vertex, two right-angled triangles \( \triangle M A B \) and \( \triangle M C D \) are inscribed in the parabola. Find the coordinates of the intersection point \( E \) of the line segments \( AB \) and \( CD \).

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