
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 shaded region shown consists of 15 unit squares and rests along the $x$-axis and the $y$-axis. The shaded region is rotated about the $y$-axis to form a solid. In cubic units, what is the volume of the resulting solid? Express your answer in simplest form in terms of $\pi$.

[asy]
defaultpen(linewidth(0.7));
fill((0,0)--(0,6)--(1,6)--(1,3)--(5,3)--(5,0)--cycle, gray(.8));
draw((0,0)--(0,6)--(1,6)--(1,3)--(5,3)--(5,0)--cycle);
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(1,3));
draw((0,4)--(1,4));
draw((0,5)--(1,5));
draw((1,0)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,3));
draw((4,0)--(4,3));

draw((5,0)--(6,0),EndArrow(4));
draw((0,0)--(-1,0),EndArrow(4));
label("$x$", (6,0), E);

draw((0,6)--(0,7),EndArrow(4));
draw((0,0)--(0,-1),EndArrow(4));
label("$y$", (0,7), N);
[/asy]

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