
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  $ ABC$  be a triangle with  $ \angle BAC \equal{} 90^\circ$ .  A circle is tangent to the sides  $ AB$  and  $ AC$  at  $ X$  and  $ Y$  respectively, such that the points on the circle diametrically opposite  $ X$  and  $ Y$  both lie on the side  $ BC$ . Given that  $ AB \equal{} 6$ , find the area of the portion of the circle that lies outside the triangle.

[asy]import olympiad;
import math;
import graph;

unitsize(20mm);
defaultpen(fontsize(8pt));

pair A = (0,0);
pair B = A + right;
pair C = A + up;

pair O = (1/3, 1/3);

pair Xprime = (1/3,2/3);
pair Yprime = (2/3,1/3);

fill(Arc(O,1/3,0,90)--Xprime--Yprime--cycle,0.7*white);

draw(A--B--C--cycle);
draw(Circle(O, 1/3));
draw((0,1/3)--(2/3,1/3));
draw((1/3,0)--(1/3,2/3));

label(" $A$ ",A, SW);
label(" $B$ ",B,  down);
label(" $C$ ",C,  left);
label(" $X$ ",(1/3,0),  down);
label(" $Y$ ",(0,1/3),  left);[/asy]

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