
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 \( x_{1}, x_{2}, x_{3} \) be non-negative real numbers that satisfy \( x_{1} + x_{2} + x_{3} = 1 \). Find the minimum and maximum values of \( \left(x_{1}+3x_{2}+5x_{3}\right)\left(x_{1}+\frac{x_{2}}{3}+\frac{x_{3}}{5}\right) \).

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