6. Consider the integral function $F ( x ) = \int _ { a } ^ { x } \frac { \cos \left( \frac { 1 } { t } \right) } { t ^ { 2 } } d t , \text{ with } x \geq a$, in which $a$ denotes a positive real parameter. Determine the largest value of $a$ so that $F \left( \frac { 2 } { \pi } \right) = - \frac { 1 } { 2 }$.