
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 $X.$ Determine the length of $XY$, given that the radius of the circle $OY = 10$.
[asy]
draw((0,0)--(10,0),black+linewidth(1));
draw((0,0)--(5,-8.66025)..(10,0)..(5,8.66025)--(0,0),black+linewidth(1));
draw((5,-8.66025)--(5,8.66025),black+linewidth(1));
label("$O$",(0,0),W);
label("$A$",(5,8.66025),N);
label("$B$",(5,-8.66025),S);
label("$X$",(5,0),NW);
label("$Y$",(10,0),E);
label("10",(0,0)--(5,8.66025),NW);
label("10",(0,0)--(5,-8.66025),SW);
[/asy]

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