
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
Given that $PQRS$ is a square and $AP = PC = 2$, find the number of square units in the area of the regular octagon.

[asy]
real x = sqrt(8);
pair A,P,C,Q,R,S,G,H;
S=(0,0); R=(4,0); Q=(4+x,x); C=(4+x,4+x);
P=(4,4+2x); A=(0,4+2x); H=(-x,4+x); G=(-x,x);
draw(A--P--C--Q--R--S--G--H--cycle);
draw((-x,0)--(4+x,0)--(4+x,4+2x)--(-x,4+2x)--cycle);
label("$P$",(-x,4+2x),NW); label("$Q$",(4+x,4+2x),NE); label("$R$",(4+x,0),SE); label("$S$",(-x,0),SW);
label("$A$",(-x,x+4),W); label("$C$",(0,4+2x),N);
[/asy]

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