
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  $p = 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}. $  For nonnegative reals  $x, y,z$  satisfying  $(x-1)^2 + (y-1)^2 + (z-1)^2 = 27,$  find the maximum value of  $x^p + y^p + z^p.$ 

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