
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
Consider a shaded region that consists of 13 unit squares along the $x$-axis and $y$-axis. It is divided into two regions. The bottom region has height 4 units, and the top region is directly above it with a narrow strip extending to the right. The shaded region is rotated about the $x$-axis to form a solid. Determine the volume of the resulting solid in cubic units, expressed in terms of $\pi$.

[asy]
defaultpen(linewidth(0.8));
fill((0,0)--(0,4)--(1,4)--(1,1)--(6,1)--(6,0)--cycle, gray(0.7));
draw((0,0)--(0,4)--(1,4)--(1,1)--(6,1)--(6,0)--cycle);
draw((0,1)--(6,1));
draw((0,2)--(1,2));
draw((0,3)--(1,3));
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
draw((4,0)--(4,1));
draw((5,0)--(5,1));

draw((6,0)--(7,0),EndArrow(4));
draw((0,0)--(-1,0),EndArrow(4));
label("$x$", (7,0), E);

draw((0,4)--(0,5),EndArrow(4));
draw((0,0)--(0,-1),EndArrow(4));
label("$y$", (0,5), N);
[/asy]

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