
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  $\mathcal P$  denote the set of planes in three-dimensional space with positive  $x$ ,  $y$ , and  $z$  intercepts summing to one. A point  $(x,y,z)$  with  $\min \{x,y,z\} > 0$  lies on exactly one plane in  $\mathcal P$ . What is the maximum possible integer value of  $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$ ?

*Proposed by Sammy Luo*

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