
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  $k$  be  a real  number. Define on the set of reals the operation  $x*y$ =  $\frac{xy}{x+y+k}$  whenever  $x+y$  does not equal  $-k$ . Let  $x_1<x_2<x_3<x_4$  be the roots of  $t^4=27(t^2+t+1)$ .suppose that  $[(x_1*x_2)*x_3]*x_4=1$ . Find all possible values of  $k$ 

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