Capturing servername and appending/propogating it to all other lines of logfile

Hello Guys,
I am very new to ELK stack. I am trying to parse below mentioned log using logstash config.
I figured out the grok pattern for last 3 lines. I need help in capturing the server name mentioned in either line1 or line 4 and append it to last 3 lines.

is this possible?

below is logfile :
XYZ V5.0; Process Startup; Server:MATLK657N1;
06-JUN-2020 12:03:31.61

H Machine:MATLK657N1,Date:20220607.
S Id:{1137-47E8-A5B0-601C86B166CD},Method:DFSList(),Start:06-JUN-2021
C Id:{1137-47E8-A5B0-671C86B166CF},Method:DFSList(),end:06-JUN-2021
S Id:{1137-47E8-A5B0-691C86B166AC},Method:DFSList(),Start:06-JUN-2021

There is an example of capturing data from one line and adding it to others here. Note the prerequisites at the end of that post.

This was super helpful. Thanks a ton!!