The Score the Final Has to Carry

A course grade is a weighted mean of four components:

• Homework — $20\%$
• Midterm 1 — $20\%$
• Midterm 2 — $20\%$
• Final exam — $40\%$

Two rules from the syllabus apply:

1. The homework component is the mean of the student's five homework scores after the single lowest homework score is dropped.
2. Weight-transfer clause: if the final exam score is strictly greater than the lower of the two midterm scores, that lower midterm is dropped from the grade entirely and its $20\%$ weight is transferred to the final exam.

Ravi's results so far:

• Homework scores: $96,\; 78,\; 91,\; 88,\; 99$
• Midterm 1: $68$
• Midterm 2: $81$

Every score is out of $100$, and the final exam score is recorded to two decimal places.

What is the lowest final exam score that gives Ravi a course grade of at least $86$?

Round your answer to two decimal places.