Problem: You are given a list of positive integers. Your task is to find the maximum sum that can be obtained by selecting a subset of these integers, subject to the following conditions:

$1.$ You can only select adjacent integers from the list.

$2.$ You cannot select two adjacent integers from the list.

$3.$ You must use at least one integer from the list.

Implement a function that takes the list of integers as input and returns the maximum sum that satisfies the given conditions. You should solve this problem using the combination of greedy method, dynamic programming, and randomized algorithm.