
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
Given that the real numbers \(x, y, z > 0\), find the minimum value of the following expressions:
(1) \(\max \left\{x, \frac{1}{y}\right\}+\max \left\{y, \frac{2}{x}\right\}\);
(2) \(\max \left\{x, \frac{1}{y}\right\}+\max \left\{y, \frac{2}{z}\right\}+\max \left\{z, \frac{3}{x}\right\}\).

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