Four Claims in a Prism

In a regular triangular prism $ABC\text{-}A_1B_1C_1$ with $AB=AA_1=1$, a point $P$ satisfies

Determine, with proof, which of the following claims are true.

1. When $\lambda=1$, the perimeter of $\triangle AB_1P$ is constant. 2. When $\mu=1$, the volume of the tetrahedron $P\text{-}A_1BC$ is constant. 3. When $\lambda=\dfrac 12$, there is exactly one point $P$ with $A_1P\perp BP$. 4. When $\mu=\dfrac 12$, there is exactly one point $P$ such that $A_1B$ is perpendicular to the plane $AB_1P$.