
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
Given $P(\cos \alpha, \sin \alpha)$, $Q(\cos \beta, \sin \beta)$, and $R(\cos \alpha, -\sin \alpha)$, if the cosine distance between $P$ and $Q$ is $\frac{1}{3}$ and between $Q$ and $R$ is $\frac{1}{2}$, calculate $\tan \alpha \cdot \tan \beta$.

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