
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 triangle  $ABC$ ,  $AB = 52$ ,  $BC = 34$  and  $CA = 50$ . We split  $BC$  into  $n$  equal segments by placing  $n-1$  new points. Among these points are the feet of the altitude, median and angle bisector from  $A$ . What is the smallest possible value of  $n$ ?

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