
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  $\triangle ABC$  be equilateral with integer side length. Point  $X$  lies on  $\overline{BC}$  strictly between  $B$  and  $C$  such that  $BX<CX$ . Let  $C'$  denote the reflection of  $C$  over the midpoint of  $\overline{AX}$ . If  $BC'=30$ , find the sum of all possible side lengths of  $\triangle ABC$ .

*Proposed by Connor Gordon*

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