
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 such that \( x_{1} + x_{2} + x_{3} = 1 \). 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}
user