
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
Circle \( A \) has its center at \( A(4, 4) \) and has a radius of 4 units. Circle \( B \) with its center at \( B(10, 4) \) has a radius of 6 units. What is the area of the gray region bound by the circles and the \( x \)-axis?
[asy]
import olympiad; size(150); defaultpen(linewidth(0.8));
xaxis(0,16,Ticks("%",1.0));
yaxis(0,8,Ticks("%",1.0));
fill((4,4)--(10,4)--(10,0)--(4,0)--cycle,gray(0.7));
filldraw(circle((4,4),4),fillpen=white);
filldraw(circle((10,4),6),fillpen=white);
dot("$A$",(4,4),S); dot("$B$",(10,4),S);
[/asy]

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