
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  $x,y$  and  $z$  be positive real numbers such that  $xy+z^2=8$ . Determine the smallest possible value of the expression   $$ \frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}. $$  

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