
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 square with side length $10$ cm is given. Points $A$ and $B$ are respectively $\frac{1}{3}$ and $\frac{2}{3}$ the length of the top and bottom sides of the square from the left corner. Lines are drawn from each corner of the square to these points, dividing the square into four triangles, two of which are congruent by symmetry. Calculate the area of the region formed by the intersecting lines within the square (excluding the triangles and the region outside).

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