
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 $f(x) = ax^3 + 3x^2 - 6ax - 11$, $g(x) = 3x^2 + 6x + 12$, and the line $m: y = kx + 9$, also $f'(-1) = 0$.

(1) Find the value of $a$;  
(2) Determine if there exists a value of $k$ such that line $m$ is both a tangent to the curve $y = f(x)$ and the curve $y = g(x)$. If it exists, find the value of $k$; if not, explain why.

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