
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 $f(x)$ have a derivative $f'(x)$ on $R$. For any real number $x$, $f(x) = 4x^{2} - f(-x)$. When $x \in (-\infty, 0)$, $f'(x) + \frac{1}{2} < 4x$. If $f(m+1) \leq f(-m) + 3m + \frac{3}{2}$, determine the range of real number $m$.

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