
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 $a$ and $b$ satisfy the inequality $|x^2 + ax + b| \leq |2x^2 - 4x - 16|$ for all $x \in \mathbb{R}$,

(1) Verify that $a = -2, b = -8$ satisfy the condition.
(2) Find all real numbers $a$ and $b$ that satisfy the condition and explain the reason.
(3) If for all $x > 2$, the inequality $x^2 + ax + b \geq (m + 2)x - m - 15$ holds, determine the range of the real number $m$.

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