
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 stated in "Elements" that "the volume $(V)$ of a sphere is directly proportional to the cube of its diameter $(D)$," which means $V = kD^3$. Euclid did not provide a value for $k$. In the 17th century, Japanese mathematicians described the method to calculate the volume of a sphere as "sphere coefficient" or "jade accumulation rate." Similarly, for a regular cylinder (a cylinder whose base is a square) and a cube, the formula $V = kD^3$ can also be used to calculate the volume (where $D$ represents the diameter of the base circle in the case of the equal-sided cylinder, and the edge length in the case of the cube). If we use this volume formula to calculate the "jade accumulation rate" for a sphere (with diameter $a$), an equal-sided cylinder (base circle diameter $a$), and a cube (with edge length $a$), denoted as $k_1$, $k_2$, and $k_3$ respectively, then the ratio $k_1 : k_2 : k_3$ is ______.

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