
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 $a > 0, b > 0$, the maximum value of the function $f(x) = |x - a| - |x + b|$ is $3$.  
$(1)$ Find the value of $a + b$;  
$(2)$ Let the function $g(x) = -x^2 - ax - b$, if for all $x \geqslant a$ we have $g(x) < f(x)$, find the range of values for $a$.

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