(1) Find the center o OF the arc CF and connect OP, of, OG. P, PI parallel CD crosses OG to I, PJ parallel AB crosses J. Then OP is perpendicular to MN, and the angle MPI=θ, so OPI= dichotomy-θ. In the triangle OPI, OI=cosθ and OJ=sinθ, so that the distances from P to AB and CD are 2-sinθ and 2-cosθ respectively, so that Mn = MP+NP = (2-COSθ)/SINθ+.

(2) The length that can cross the corner must be the minimum of the above formula, but this length is already the maximum length that can cross the corner, so there is something wrong with the description of the topic, because the minimum length that can cross the corner must be 0.

Leaving this aside, let's find the minimum value of the above expression. Because the angles are symmetrical, for example, the length of 50 and 40 angles must be the same. So it can be considered that the minimum value should be 45. That is, the minimum value is four times the root number 2-2.

Of course, this conclusion needs to be proved.