
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 $a$, $b$, and $c$ be real numbers such that $3a$, $4b$, and $5c$ form a geometric sequence, and $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ form an arithmetic sequence. Find the value of $\frac{a}{c} + \frac{c}{a}$.

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