<Tri-2025-2183>, hint

Suppose AM cuts the circumcircle of ABC at O and I is incenter of ABC.

Since <OBI=<OIB=<OCI, O is the center of circumcircle of BIC.

Hence <BPM+<APC=180

Since AO is the diameter of the circumcircle of ABC, <ABO=<ACO

circle BIC is tangent to AC, AB and so <ACP=<PBC