
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  $x,y$  be complex numbers such that  $\dfrac{x^2+y^2}{x+y}=4$  and  $\dfrac{x^4+y^4}{x^3+y^3}=2$ .  Find all possible values of  $\dfrac{x^6+y^6}{x^5+y^5}$ .

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