
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  $ ABCD$  be a unit square (that is, the labels  $ A, B, C, D$  appear in that order around the square). Let  $ X$  be a point outside of the square such that the distance from  $ X$  to  $ AC$  is equal to the distance from  $ X$  to  $ BD$ , and also that  $ AX \equal{} \frac {\sqrt {2}}{2}$ .  Determine the value of  $ CX^2$ .

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