Verify the inequality! Left in sandbox for at least 3 days.
I want to verify if this inequality is true:
for \$n\geq4\$, if \$a_1,a_2,a_3,\dots,a_n\in R_+\cup\{0\}\$ and \$\sum_{i=1}^na_i=1\$, then \$a_1a_2+a_2a_3+a_3a_4+\dots+a_{n-1}a_n+a_na_1\leq\frac{1}{4}\$.
Challenge
Write a piece of program which takes an integer n as input. It does the following:

Generate a random array a which consists of n non-negative reals. The sum of all elements should be 1.


By saying random, I mean, every array satisfiying the requirements in 2 should have a non-zero probability of occurrence. It don't need to be uniform. See this related post.


Calculate a[0]a[1]+a[1]a[2]+a[2]a[3]+...+a[n-2]a[n-1]+a[n-1]a[0].

Output the sum and the array a.



For I/O forms see this post.

Rules
(Sorry for the late edit...) All numbers should be rounded to at least \$10^{-4}\$.
Standard loopholes should be forbidden.
Example
The following code is an ungolfed Python code for this challenge, using library numpy. (For discussion about using libraries, see This Link.)
import numpy as np


def inequality(n):
    if n &lt; 4:
       raise Exception
    a = np.random.rand(n)
    sum_a = 0
    for i in range(n):
        sum_a += a[i]
    for i in range(n):
        a[i] /= sum_a
    sum_prod = 0
    for i in range(n):
       sum_prod += a[i % n] * a[(i + 1) % n]
    print(a)
    return sum_prod, a

Tip
You could assume that input n is a positive integer greater than 3.

Your score is the bytes in your code. The one with the least score wins.
APL(Dyalog Unicode), 21 bytes SBCS
(+/⊢×1∘⌽)⎕←(⊢÷+/)?⎕⍴0

Try it on APLgolf!
A tradfn submission which prints the list and returns the sum.
                  ⎕⍴0 ⍝ create a vector with input-many 0's
                 ?    ⍝ for each 0, get a random number between 0 and 1
            ⊢÷        ⍝ divide each number by
              +/      ⍝ the sum of all numbers
         ⎕←           ⍝ print the resulting vector
   ⊢×                 ⍝ multiply the vector element-wise
     1∘⌽              ⍝ with the vector rotated to the left by 1
 +/                   ⍝ take the sum of all products

If appending the sum to the list is fine as output, this could be 20 bytes: x,+/x×1⌽x←(⊢÷+/)?⎕⍴0