
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
A group of three gears $G_1$, $G_2$, and $G_3$ is aligned such that each pair meshes directly with each other. Gear $G_1$ has $2x$ teeth, Gear $G_2$ has $3y$ teeth, and Gear $G_3$ has $4z$ teeth where $x$, $y$, and $z$ are positive integers. If the gears mesh properly with the same spacing and tooth size, what is the relationship of their angular speeds, $\omega_1$, $\omega_2$, and $\omega_3$, respectively?
A) $\frac{1}{6yx} : \frac{1}{2xz} : \frac{1}{3yz}$
B) $\frac{2z}{x} : \frac{4z}{3y} : 1$
C) $\frac{2z}{3y} : \frac{4z}{x} : 1$
D) $\frac{3y}{2x} : \frac{x}{z} : 1$
E) $\frac{x}{2z} : \frac{y}{3z} : 1$

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