
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 \(0 < \alpha < \beta < \gamma \in \mathbb{R}\), the reals. Let 
\[ K = \{ (x, y, z) : x^\beta + y^\beta + z^\beta = 1, \text{ and } x, y, z \geq 0 \} \in \mathbb{R}^3. \]
Define \( f : K \to \mathbb{R} \) by 
\[ f(x, y, z) = x^\alpha + y^\beta + z^\gamma. \]
At what points of \(K\) does \(f\) assume its maximum and minimum values?

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