
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

Find all pairs \((\alpha, \beta)\) of positive real numbers with the following properties:

(a) For all positive real numbers \(x, y, z, w\) it holds that
\[ 
x + y^2 + z^3 + w^6 \geq \alpha (xyzw)^\beta 
\]

(b) There is a quadruple \((x, y, z, w)\) of positive real numbers such that equality holds in (a).

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