
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 region consisting of 13 unit squares placed as below. The region is rested along the $x$-axis and the $y$-axis. The part of the region from $x=0$ to $x=1$ and $y=0$ to $y=6$ is rotated about the $x$-axis. The part from $x=1$ to $x=4$ and $y=0$ to $y=2$ is rotated about the $y$-axis. Calculate the volume of the resulting solid in cubic units, expressed in terms of $\pi$.

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

draw((4,0)--(5,0),EndArrow(4));
draw((0,0)--(-2,0),EndArrow(4));
label("$x$", (5,0), 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