B777 Captain reported during cruise flight experiencing left engine anti-ice failure resulting in avoiding icing conditions as much as possible and landing safely at their destination airport.

Narrative

On Day 0; we were enroute from ZZZZ to ZZZ. At approximately XA:20 and 85 miles to fix ZZZZZ; we received a Anti-Ice Engine Left caution EICAS message. We completed the non-normal checklist and the anti-ice valve remained inoperative; the checklist directed us to avoid icing conditions. We contacted Dispatch and conferenced in Maintenance for a three-way call. Maintenance did not have any procedures or suggestions to fix the engine anti ice valve.There were no suitable airports or routes for us to land and completely avoid icing conditions We made a joint decision with Dispatch to proceed to ZZZ. We [requested priority handling]; spent as little time as possible in icing conditions and landed successfully in ZZZ. Cause: Faulty engine anti ice valve.