
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 real numbers \( x_{1}, x_{2}, \cdots, x_{10} \) that satisfy
$$
\sum_{i=1}^{10}\left|x_{i}-1\right| \leq 4, \quad \sum_{i=1}^{10}\left|x_{i}-2\right| \leq 6,
$$

find the average value \(\bar{x}\) of \( x_{1}, x_{2}, \cdots, x_{10} \).

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