
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 $f(x)$ has a derivative $f'(x)$ on $\mathbb{R}$ and satisfies $f(x)=4x^2-f(-x)$, and $f'(x) + \frac{1}{2} < 4x$ for $x \in (-\infty,0)$, find the range of the real number $m$ that satisfies $f(m+1) \leqslant f(-m)+4m+2$.

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