
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 a system of three meshed gears $A$, $B$, and $C$ with $p$, $q$, and $r$ teeth respectively, it is given that the angular speed of gear $A$ is directly twice that of what would be expected from a simple teeth count calculation with uniform linear speed at the meshing point. Find the proportion of angular speeds for gears $A$, $B$, and $C$.
A) $qr : pr : pq$
B) $pr : qr : 2pq$
C) $2qr : pr : pq$
D) $pq : qr : 2pr$

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