
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
Let $O$ be the origin, and let $(d,e,f)$ be a fixed point. A plane passes through $(d,e,f)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at points $D,$ $E,$ and $F,$ respectively, all distinct from $O.$ Let $(u,v,w)$ be the center of the sphere passing through $D,$ $E,$ $F,$ and $O.$ Find
\[\frac{d}{u} + \frac{e}{v} + \frac{f}{w}.\]

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