The half$-$precision floating$-$point system is often used in machine learning applications; to train machine learning algorithms, a large amount of calculation is necessary, but can be done without a high level of precision. This floating$-$point system uses only five bits for the exponent and ten bits for the significand, which can make it significantly faster than double precision.

Half$-$precision numbers are represented as

with

and exponent

$.$

What is the largest value of

such that all integers in the range

are exactly representable in this floating point system? Your answer should be a decimal value.