
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
What is the largest \( n \) for which we can find \( n + 4 \) points in the plane, \( A, B, C, D, X_1, \ldots, X_n \), so that \( AB \neq CD \), but for each \( i \) the two triangles \( ABX_i \) and \( CDX_i \) are congruent?

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