BSW phenomenon for near-fine-tuned particles with external force: general classification of scenarios

If two particles moving towards a black hole collide in the vicinity of the horizon, the energy $E_{c.m.}$ in the center of mass frame can grow indefinitely if one of particles is fine-tuned. This is the Ba\~{n}ados, Silk and West (BSW) effect. One of objections against this effect consists in that for some types of a horizon fine-tuned particles cannot reach the horizon. However, this difficulty can be overcome if instead of exact fine-tuning, one of particle is nearly fine-tuned, with the value of small detuning being adjusted to the distance to the horizon. Such particles are called near-fine-tuned. We give classification of such particles and describe possible high energy scenarios of collision in which they participate. We analyze the ranges of possible motion for each type of particle and determine under which condition such particles can reach the horizon. We analyze collision energy $E_{c.m.}\,$and determine under which conditions it may grow indefinitely. We also include into consideration the forces acting on particles and find when the BSW effect with nearly-fine-tuned particles is possible with finite forces. We demonstrate that the BSW effect with particles under discussion is consistent with the principle of kinematic censorship. According to this principle, $E_{c.m.}\,$cannot be literally infinite in any event of collision (if no singularity is present), although it can be made as large as one likes.