
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
For which natural numbers $n \geq 2$ does the inequality $x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2} \geq p\left(x_{1} x_{2}+x_{2} x_{3}+\ldots+x_{n-1} x_{n}\right)$ hold for any real numbers $x_{1}, x_{2}, \ldots, x_{n}$, if
a) $p=1$;

b) $p=4 / 3$;

c) $p=6 / 5$?

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