
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
The nine points of this grid are equally spaced horizontally and vertically. The distance between two neighboring points is 1 unit. What is the area, in square units, of the region where the two quadrilaterals overlap?

[asy]
size(80);
dot((0,0)); dot((0,1));dot((0,2));dot((1,0));dot((1,1));dot((1,2));dot((2,0));dot((2,1));dot((2,2));
draw((0,1)--(1,2)--(2,1)--(1,0)--cycle, linewidth(0.6));
draw((0,0)--(2,2)--(2,0)--(0,2)--cycle, linewidth(0.6));
[/asy]

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