
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 $P$ and $Q$ are the midpoints of two adjacent sides of the square. What is the area of the shaded region?

[asy]
draw((0,0)--(12,0));
draw((12,0)--(12,12));
draw((12,12)--(0,12));
draw((0,0)--(0,12));
draw((0,0)--(6,12));
draw((6,12)--(12,0));
draw((0,12)--(6,0));
draw((6,0)--(12,12));
fill((6,0)--(9,6)--(6,12)--(3,6)--cycle,gray);
label("P",(6,12),N);
label("Q",(6,0),S);
[/asy]

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