
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 an equilateral triangle with side length  $1$ .  Say that a point  $X$  on side  $\overline{BC}$  is *balanced* if there exists a point  $Y$  on side  $\overline{AC}$  and a point  $Z$  on side  $\overline{AB}$  such that the triangle  $XYZ$  is a right isosceles triangle with  $XY = XZ$ .  Find with proof the length of the set of all balanced points on side  $\overline{BC}$ . 

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