
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
(1) Given a hyperbola $C$ with its center at the origin and its right focus at $(2,0)$, and the length of its real axis is 2. Find the equation of the hyperbola $C$.
(2) Suppose the directrix of the parabola $y^2 = mx$ ($m \neq 0$) is at a distance of 2 from the line $x = -1$. Find the equation of the parabola.

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