
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. Points $A$ and $B$ are at the third and two-thirds points along the top and bottom sides of the square, respectively. What is the area of the shaded region, assuming it is shaped similarly to the earlier problem?

[asy]
draw((0,0)--(15,0));
draw((15,0)--(15,15));
draw((15,15)--(0,15));
draw((0,0)--(0,15));
draw((0,0)--(7.5,15));
draw((7.5,15)--(15,0));
draw((0,15)--(7.5,0));
draw((7.5,0)--(15,15));
fill((7.5,0)--(11.25,7.5)--(7.5,15)--(3.75,7.5)--cycle,gray);
label("A",(7.5,15),N);
label("B",(7.5,0),S);
[/asy]

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