
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
Given a set of data \( x_{1}, x_{2}, \cdots, x_{6} \) with variance
\[ 
S^{2} = \frac{1}{6} \left( x_{1}^{2} + x_{2}^{2} + \cdots + x_{6}^{2} - 24 \right), 
\]
find the mean of the data \( x_{1} + 1, x_{2} + 1, \cdots, x_{6} + 1 \).

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