
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 line with a slope of $1$ intersects a hyperbola with an eccentricity of $\sqrt{3}$ at points $P$ and $Q$. The line intersects the $y$-axis at point $R$, and $\overrightarrow{OP} \cdot \overrightarrow{OQ} = -3$, $\overrightarrow{PR} = 3\overrightarrow{RQ}$. Find the equations of the line and the hyperbola.

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