
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 such that points $A$ and $B$ are on two opposite sides of the square at one-third and two-thirds the length of the sides, respectively. What is the area of the new shaded region formed by connecting points $A$, $B$, and their reflections across the square's diagonal?

[asy]
draw((0,0)--(15,0));
draw((15,0)--(15,15));
draw((15,15)--(0,15));
draw((0,15)--(0,0));
draw((0,5)--(15,10));
draw((15,5)--(0,10));
fill((7.5,2.5)--(7.5,12.5)--(5,7.5)--(10,7.5)--cycle,gray);
label("A",(0,5),W);
label("B",(15,10),E);
[/asy]

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