
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
In the 3rd century BC, the ancient Greek mathematician Euclid proposed in "Elements" that "the volume (V) of a sphere is directly proportional to the cube of its diameter (D)," which is $V=kD^3$, but Euclid did not provide the value of $k$. In the 17th century, Japanese mathematicians were still unfamiliar with the method of calculating the volume of a sphere, and they referred to the constant $k$ in the volume formula $V=kD^3$ as the "rate of sphere" or "jade rate". Similarly, for a right circular cylinder (whose cross-section is a square) and a cube, the volume formula $V=kD^3$ can also be used (in the right circular cylinder, $D$ represents the diameter of the base circle; in the cube, $D$ represents the edge length). Assuming the "jade rate" for a sphere (with diameter $a$), a right circular cylinder (with base circle diameter $a$), and a cube (with edge length $a$) are $k_1$, $k_2$, and $k_3$ respectively, then the ratio $k_1:k_2:k_3$ is $\boxed{\frac{\pi}{6}:\frac{\pi}{4}:1}$.

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