Townhouse water damage through shared wall – Who’s liable?

I'd like to get a local perspective on this. I live in a townhouse with shared walls on both sides. This morning I woke up to a pool of water in my garage that appeared to be coming from a shared wall. We checked the guest bedroom behind the garage and noticed the carpet was wet and the drywall was also wet. Turns out my neighbors toilet was leaking. I called our insurance to walk through the situation and find out the best course of action but tbh they didn’t seem very knowledgeable so it wasn’t of much help.

My question – Is my neighbor liable for damages to my property due to a leak coming from a pipe in their home? My homeowners deductible is $3.5k so I’d rather not go that avenue since the damages will probably be in that ballpark. I imagine the drywall in the guest bedroom will need to be torn out, carpet torn out and mold remediation will need to take place. I still don't know if the source is from toilet waste water or a pressurized supply line. Should I file a claim with my Home owners insurance policy and take them to small claims court to pursue the deductible? I know this might be a waste of time but their radio silence about my inquires on the source of the leak has made me want to waste their time in court. Any suggestions or ideas on how to proceed?