
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
Determine the values of the positive integer \( n \) for which the following system of equations has a solution in positive integers \( x_{1}, x_{2}, \ldots, x_{n} \). Find all solutions for each such \( n \).

\[
\begin{aligned}
x_{1} + x_{2} + \cdots + x_{n} &= 16 \\
\frac{1}{x_{1}} + \frac{1}{x_{2}} + \cdots + \frac{1}{x_{n}} &= 1
\end{aligned}
\]

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