
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 \(x_{1}, x_{2}, x_{3}, \cdots, x_{9}\) be positive integers such that \(x_{1} < x_{2} < \cdots < x_{9}\) and \(x_{1} + x_{2} + \cdots + x_{9} = 220\). Determine the minimum value of \(x_{9} - x_{1}\) when the value of \(x_{1} + x_{2} + \cdots + x_{5}\) is maximized.

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