
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 a given integer  $n\geq 3$ , determine the range of values for the expression
\[ E_n(x_1,x_2,\ldots,x_n) := \dfrac {x_1} {x_2} + \dfrac {x_2} {x_3} + \cdots + \dfrac {x_{n-1}} {x_n} + \dfrac {x_n} {x_1}\]
over real numbers  $x_1,x_2,\ldots,x_n \geq 1$  satisfying  $|x_k - x_{k+1}| \leq 1$  for all  $1\leq k \leq n-1$ .
Do also determine when the extremal values are achieved.

(Dan Schwarz)

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