
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
Consider a circle of radius 1 with center $O$ down in the diagram. Points $A$, $B$, $C$, and $D$ lie on the circle such that $\angle AOB = 120^\circ$, $\angle BOC = 60^\circ$, and $\angle COD = 180^\circ$. A point $X$ lies on the minor arc $\overarc{AC}$. If $\angle AXB = 90^\circ$, find the length of $AX$.

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