
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 the function \( f(x) = x^2 + a|x| + x \), where \( x \in \mathbf{R} \) and \( a \in \mathbf{R} \):

1. If \( x_1 \) and \( x_2 \in (0, +\infty) \), compare \( \frac{1}{2} [ f(x_1) + f(x_2) ] \) and \( f\left(\frac{x_1 + x_2}{2}\right) \).
2. If \( x_1 \) and \( x_2 \in \mathbf{R} \), and \( \frac{1}{2} [ f(x_1) + f(x_2) ] \) is consistent with \( f\left(\frac{x_1 + x_2}{2}\right) \) as in (1), find the range of the value of \( a \).

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