
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 $m:y=kx+9$, and $f'(-1)=0$.

(1) Find the value of $a$.

(2) Is there a value of $k$ such that line $m$ is both a tangent line 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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