
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 \( x_{1}, \ldots, x_{n} \) be real numbers strictly greater than 1. Consider the number of real numbers of the form \( \varepsilon_{1} x_{1} + \cdots + \varepsilon_{n} x_{n} \) where \( \varepsilon_{i} \in \{0, 1\} \) that lie within a given interval of length 1. What is the maximum number that can be obtained this way?

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