
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 $12$-cm-by-$12$-cm square is partitioned similarly to the original problem, but points $A$ and $B$ are located at one-third and two-thirds of the length of two opposite sides of the square, respectively. Calculate the area of the similarly shaded region formed by the intersections of lines drawn from each corner to $A$ and $B$.

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