
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
Points  $A(0,0)$  and  $B(1,1)$  are located on the parabola  $y=x^2$ .  A third point  $C$  is positioned on this parabola between  $A$  and  $B$  such that  $AC=CB=r$ .  What is  $r^2$ ?

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