
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
In the diagram, $AOB$ is a sector of a circle with $\angle AOB=120^\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY$? Assume the radius $OA=OB=12$ units. [asy]
draw((0,0)--(12,0),black+linewidth(1));
draw((0,0)--(-6,10.3923)..(12,0)..(-6,-10.3923)--(0,0),black+linewidth(1));
draw((-6,10.3923)--(-6,-10.3923),black+linewidth(1));
label("$O$",(0,0),W);
label("$A$",(-6,10.3923),N);
label("$B$",(-6,-10.3923),S);
label("$X$",(-6,0),NW);
label("$Y$",(12,0),E);
label("12",(0,0)--(-6,10.3923),NW);
label("12",(0,0)--(-6,-10.3923),SW);
[/asy]

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