Make regex match a pattern repeatedly

I want to repetedly capture a certain pattern. My pattern is #|X|#YYY, where X is any single letter, and YYY is any string, and the # and | characters are hardcoded separator characters.

For example, this is my Input String: #|m|#this is string #1#|m|#this is string #2#|m|#this is string #3

This is my desired output:

Match:
 - Group 1: m
 - Group 2: this is string #1
Match:
 - Group 1: m
 - Group 2: this is string #2
Match:
 - Group 1: m
 - Group 2: this is string #3

I tried this:

const pattern = /#|(w+)|#(.*)/g;
const input = "#|m|#this is string #1#|m|#this is string #2#|m|#this is string #3";

let match;
while ((match = pattern.exec(input)) !== null) {
  const group1 = match[1];
  const group2 = match[2];
  console.log("Match:");
  console.log("- Group 1:", group1);
  console.log("- Group 2:", group2);
}

but it seems to capture too gredily and only outputs one match:

Match:
 - Group 1: m
 - Group 2: this is string #1#|m|#this is string #2#|m|#this is string #3

When I try to make the pattern ungreedy, I get 3 matches as desired, but group 2 is empty for some reason.

Ungreedy attempt: (notice the ?)

const pattern = /#|(w+)|#(.*?)/g;

Output:

Match:
 - Group 1: m
 - Group 2: 
Match:
 - Group 1: m
 - Group 2: 
Match:
 - Group 1: m
 - Group 2: 

What am I doing wrong? How can I make it match the whole string until the next occurence to get my desired output?