
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, three gears $P$, $Q$, and $R$ are connected, where gears $P$ and $Q$ interlock directly and gear $R$ is connected to gear $Q$ through a secondary mechanism that introduces a power transmission efficiency of 90%. The number of teeth on gears $P$, $Q$, and $R$ are $a$, $b$, and $c$ respectively. If $P$, $Q$, and $R$ have the same size and regularly spaced teeth, which proportion correctly represents the angular speeds $\omega_P$, $\omega_Q$, and $\omega_R$ in revolutions per minute respectively?
A) $b : a : 0.9a$
B) $a : b : 0.9c$
C) $c : b : a$
D) $c : a : b$

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