
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 \( x, y, z \) be real numbers such that \( 3x, 4y, 5z \) form a geometric sequence and \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) form an arithmetic sequence. Find the value of \( \frac{x}{z} + \frac{z}{x} \).

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