
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
A $10$-cm-by-$10$-cm square is partitioned as shown. Point $A$ is located one-quarter the distance down from the top on the left side of the square, and point $B$ is the midpoint of the right side of the square. The points are connected to the opposite corners, dividing the square into several regions. Calculate the area of the shaded region formed around the center.

[asy]
draw((0,0)--(12,0));
draw((12,0)--(12,12));
draw((12,12)--(0,12));
draw((0,0)--(0,12));
draw((0,9)--(12,0));
draw((0,9)--(6,12));
draw((6,12)--(12,0));
draw((6,0)--(12,12));
fill((6,0)--(9,6)--(6,12)--(3,6)--cycle,gray);
label("A",(0,9),W);
label("B",(12,6),E);
[/asy]

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