
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
Find the greatest integer  $n$ , such that there are  $n+4$  points  $A$ ,  $B$ ,  $C$ ,  $D$ ,  $X_1,\dots,~X_n$  in the plane with  $AB\ne CD$  that satisfy the following condition: for each  $i=1,2,\dots,n$  triangles  $ABX_i$  and  $CDX_i$  are equal.

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