
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
A function \( f(x) \) defined on the set \( A \) satisfies the condition: For any \( x_{1}, x_{2} \in A \),
\[ f\left(\frac{x_{1}+x_{2}}{2}\right) \leq \frac{1}{2}\left[f\left(x_{1}\right)+f\left(x_{2}\right)\right], \]
then \( f(x) \) is called a concave function on \( A \).

(1) Determine whether \( f(x)=3 x^{2}+x \) is a concave function on \( \mathbf{R} \).

(2) If the function \( f(x)=m x^{2}+x \) is a concave function on \( \mathbf{R} \), find the range of the real number \( m \).

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