Group Sum with Special Constraints

Background

Congcong has been studying how to select specific combinations from a set of numbers. He encountered an interesting problem that requires your help to solve.

Problem Description

Given an array of integers $nums$ and a target sum $target$, determine if it's possible to choose a group of some of the integers from the array such that their sum equals $target$, with these additional constraints:

1. All multiples of $5$ in the array must be included in the group.
2. If the value immediately following a multiple of $5$ is $1$, it must not be chosen. (No loops needed.)

Input Format

The input consists of a single line. This line contains an integer $start\_index$, a list of integers $nums$ enclosed in square brackets, and an integer $target$. In this problem, $start\_index$ will always be $0$.

start_index [num1 num2 ... numK] target

Output Format

Output to standard output in the following format.

Output true if such a group exists; otherwise, output false.

Sample

0 [2 5 10 4] 19

true

0 [2 5 10 4] 17

true

0 [2 5 10 4] 12

false

Sample Explanation

Sample 1: For input 0 [2 5 10 4] 19, the multiples of $5$ in the array are $5$ and $10$, which must be included. Their sum is $15$. The target sum is $19$, so $4$ more is needed. Choosing $4$ from the remaining elements $2, 4$ achieves the target. The chosen group is $5, 10, 4$, with a sum of $19$. Thus, true is output.

Sample 2: For input 0 [2 5 10 4] 17, the multiples of $5$ in the array are $5$ and $10$, which must be included. Their sum is $15$. The target sum is $17$, so $2$ more is needed. Choosing $2$ from the remaining elements $2, 4$ achieves the target. The chosen group is $5, 10, 2$, with a sum of $17$. Thus, true is output.

Sample 3: For input 0 [2 5 10 4] 12, the multiples of $5$ in the array are $5$ and $10$, which must be included. Their sum is $15$. Since $15$ is already greater than the target sum $12$, it's impossible to reach the target by adding non-negative numbers. Thus, false is output.

Constraints

Time limit: $1$ second, Memory limit: $1024$ KiB for each test case.