
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
Let  $ABC$  be a triangle with  $AB=3$  and  $AC=4$ . It is given that there does not exist a point  $D$ , different from  $A$  and not lying on line  $BC$ , such that the Euler line of  $ABC$  coincides with the Euler line of  $DBC$ . The square of the product of all possible lengths of  $BC$  can be expressed in the form  $m+n\sqrt p$ , where  $m$ ,  $n$ , and  $p$  are positive integers and  $p$  is not divisible by the square of any prime. Find  $100m+10n+p$ .

Note: For this problem, consider every line passing through the center of an equilateral triangle to be an Euler line of the equilateral triangle. Hence, if  $D$  is chosen such that  $DBC$  is an equilateral triangle and the Euler line of  $ABC$  passes through the center of  $DBC$ , then consider the Euler line of  $ABC$  to coincide with "the" Euler line of  $DBC$ .

*Proposed by Michael Ren*

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