
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
The center of a sphere \( Q \) has coordinates \( (a, b, c) \), and the radius of the sphere is \( R \). Denoting the variable radius vector by \( \vec{r} \), and the radius vector of point \( Q \) by \( \vec{q} = \overrightarrow{OQ} \), derive the vector equation of the sphere as the set of points that are at a distance \( R \) from point \( Q \). From this, obtain the equation of the sphere in coordinates (analogous to the equation of a circle in coordinates on a plane):

$$
(x-a)^{2} + (y-b)^{2} + (z-c)^{2} = R^{2}
$$

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