
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 2 is rotated about one vertex by an angle of $\alpha = 60^\circ$ where $\cos \alpha = \frac{1}{2}$. Find the area of the shaded region that is common to both squares.

[asy]
unitsize(1.5 cm);

pair A, B, C, D, Bp, Cp, Dp, P;

A = (0,0);
B = (-2,0);
C = (-2,-2);
D = (0,-2);
Bp = rotate(60)*(B);
Cp = rotate(60)*(C);
Dp = rotate(60)*(D);
P = extension(C,D,Bp,Cp);

fill(A--Bp--P--D--cycle,gray(0.7));
draw(A--B---C--D--cycle);
draw(A--Bp--Cp--Dp--cycle);

label("$\alpha$", A + (-0.5,-0.2));
[/asy]

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