
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
Find the largest positive real number  $p$  (if it exists) such that the inequality
\[x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)\]
is satisfied for all real numbers  $x_i$ , and  $(a) n = 2; (b) n = 5.$ Find the largest positive real number  $p$  (if it exists) such that the inequality holds for all real numbers  $x_i$  and all natural numbers  $n, n \ge 2.$ 

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