
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 $f(x) = ax^2 + 1$ ($a > 0$), and $g(x) = x^3 + bx$.  
(1) If the curve $y = f(x)$ and the curve $y = g(x)$ have a common tangent line at their intersection point $(1, c)$, find the values of $a$ and $b$;  
(2) When $a = 3$ and $b = -9$, the maximum value of the function $f(x) + g(x)$ in the interval $[k, 2]$ is 28, find the range of $k$.

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