
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
If $x, y, z > 0$, find the smallest possible value of
\[
\left\lfloor{\frac{2x+y}{z}}\right\rfloor + \left\lfloor{\frac{2y+z}{x}}\right\rfloor + \left\lfloor{\frac{2z+x}{y}}\right\rfloor + \left\lfloor{\frac{x+y+z}{x+y}}\right\rfloor.
\]

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