
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
If \(\alpha, \beta\) are the roots of \(x^{2} - 10x + 20 = 0\), and \(p = \alpha^{2} + \beta^{2}\), find \(p\).

The perimeter of an equilateral triangle is \(p\). If its area is \(k \sqrt{3}\), find \(k\).

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