
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
(a) If \( y = x + \frac{1}{x} \), calculate the expressions \( x^{2} + \frac{1}{x^{2}} \) and \( x^{3} + \frac{1}{x^{3}} \) in terms of \( y \).

(b) Determine all real roots of the equation \( x^{2} - 5x + 8 - \frac{5}{x} + \frac{1}{x^{2}} = 0 \).

(c) Determine all real roots of \( x^{4} - 5x^{3} + 8x^{2} - 5x + 1 = 0 \).

(d) Determine all real roots of \( x^{6} - 2x^{5} - 5x^{4} + 12x^{3} - 5x^{2} - 2x + 1 = 0 \).

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