
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 as shown. Points $A$ and $C$ are the midpoints of the top side and the right side of the square, respectively, and point $B$ is three-quarters down on the left side of the square. Calculate the area of the shaded region which is enclosed by the lines $AB$, $BC$, and the diagonal $AC$.

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

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