Grouping Sum with Specific Conditions

Background

Congcong has recently been studying how to cleverly group a set of numbers. He wants to know if it's possible to divide these numbers into two groups such that the sum of the two groups is the same, while satisfying specific conditions.

Problem Description

Given an array of integers, determine if it's possible to divide these integers into two groups such that the sum of the two groups is the same. Additionally, the following constraints must be met: all values that are multiples of $5$ must be in one group, and all values that are multiples of $3$ (and not multiples of $5$) must be in the other. (No loops needed.)

Input Format

The input is given from standard input in the following format.

The input consists of a single line containing space-separated integers, representing the elements of an array.

Output Format

Output to standard output in the following format.

Output true if it's possible to divide the integers into two groups with equal sums according to the requirements; otherwise, output false.

Sample

1 1

true

1 1 1

false

2 4 2

true

Sample Explanation

Sample 1: Input 1 1.

• There are no multiples of $5$ or $3$ in the array.
• The total sum is $2$, and the target sum for each group is $1$.
• We can put the first $1$ into group 1 and the second $1$ into group 2. Both groups have a sum of $1$, which are equal. So, the output is true.

Sample 2: Input 1 1 1.

• There are no multiples of $5$ or $3$ in the array.
• The total sum is $3$. Since the total sum is odd, it cannot be divided into two groups with equal sums. So, the output is false.

Sample 3: Input 2 4 2.

• There are no multiples of $5$ or $3$ in the array.
• The total sum is $8$, and the target sum for each group is $4$.
• We can put the first $2$ and the third $2$ into group 1, making its sum $4$; and put $4$ into group 2, making its sum $4$. The sums of both groups are equal. So, the output is true.

Constraints

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