How do I increase the number of rings before the answering machine picks up, it is currently 4 rings, and I would like it to be 6

The CS6419 isn't an answering machine. The phone will ring until you answer, an answering machine or voicemail system picks up, or the caller hangs up.

To change the ring count on an answering machine or voicemail system, refer to the documentation for that system. If you require further assistance, please add a comment to this question and identify the answering machine or voicemail system.

Are you sharing your home phone line with your fax? If so, do you have a voicemail or an answering machine connected in the same line?

If you have an answering machine connected to the same line as the fax, set the answering machine to answer incoming calls before the printer does. Increase the number of rings by a factor of 3 over the answering machine. For example, if the answering machine is set to pick up on 4 rings, set the printer to pick up on 7 rings.

If you have a voicemail feature in the same line as the fax, you may subscribe from your phone service provider for a distinctive ring pattern feature. You can program your printer to have a distinctive ring pattern and phone number for incoming faxes.

It seems that you're sharing your home phone line with your Lexmark fax machine. You can adjust the "pick-up" ring from the printer operator panel by pressing the Setup button > Fax Setup > Ringing and Answering > Pick Up On > select your preferred number of rings before the fax answers.
Please note:
If you have an answering machine connected to the same line as the fax, set the answering machine to answer incoming calls before the printer does. Increase the number of rings by a factor of 3 over the answering machine. For example, if the answering machine is set to pick up on 4 rings, set the printer to pick up on 7 rings.

If you have a voicemail feature in the same line as the fax, you may subscribe from your phone service provider for a distinctive ring pattern feature. You can program your printer to have a distinctive ring pattern and phone number for incoming faxes.

Should you still need assistance, please leave a comment to my response and I'll get back to you. Should this resolves your problem, your rating will be greatly appreciated.

Are you sharing your home phone line with your fax? Do you have an answering machine or a voicemail feature in the same line?
If you have an answering machine connected to the same line as the fax, set the answering machine to answer incoming calls before the printer does. Increase the number of rings by a factor of 3 over the answering machine. For example, if the answering machine is set to pick up on 4 rings, set the printer to pick up on 7 rings.

If you have a voicemail feature in the same line as the fax, you may want to subscribe from your phone service provider for a distinctive ring pattern feature. You can program your printer to have a distinctive ring pattern and phone number for incoming faxes.

If you have an answering machine connected to the same line as the fax, set the answering machine to answer incoming calls before the printer does. Increase the number of rings by a factor of 3 over the answering machine. For example, if the answering machine is set to pick up on 4 rings, set the printer to pick up on 7 rings.

If you have a voicemail feature in the same line as the fax, you may subscribe from your phone service provider for a distinctive ring pattern feature. You can program your printer to have a distinctive ring pattern and phone number for incoming faxes.

The 25839 isn't an answering machine. If you have a separate answering machine connected to your phone line, consult its documentation. If you subscribe to a voicemail system offered by your phone company, contact them for instructions.

Change the answer machine rings to answer number to less than the fax number of rings to answer or increase the number of rings to answer on the fax machine. Would need exact model number of phone system to give more detail on programming steps. Info should be in manuals of phone system and fax machine.

Press MENU, then enter #06, then press the number of rings before the fax picks up by pressing 1,2,3, or 4 (4 being the maximum number of rings)
Press SET, then press MENU again to exit. Best of luck!
Greg

Press
function
64
set

look at the printout
look for number of rings

to set the parameter
press
function 74
set

the default setting is 5 rings. if you need to change that:
set the ring to answer to higher number, however, that will put the machine in manual answering of fax.
the better setting is to rest the phone or the answer machine to a lower number of rings.