
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
Let  $a$ ,  $b$ ,  $c$  be the three roots of the equation  $x^3-(k+1)x^2+kx+12=0$ , where  $k$  is a real number. If  $(a-2)^3+(b-2)^3+(c-2)^3=-18$ , find the value of  $k$ .

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