
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 given diagram, $\|\overrightarrow{OA}\| = 1$, $\|\overrightarrow{OB}\| = 1$, and $\|\overrightarrow{OC}\| = \sqrt{3}$. Also, $\tan \angle AOC = 3$ and $\angle BOC = 60^\circ$. There exist constants $p$ and $q$ such that
\[\overrightarrow{OC} = p \overrightarrow{OA} + q \overrightarrow{OB}.\]
Determine the ordered pair $(p, q)$.

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