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For a portion of national highway where a descending gradient of $1$ in $25$ meets with an ascending gradient of $1$ in $20$, a valley curve needs to be designed for a vehicle travelling at $90$ kmph based on the following conditions.

- headlight sight distance equal to the stopping sight distance (SSD) of a level terrain considering length of valley curve > SSD.
- comfort condition with allowable rate of change of centrifugal acceleration acceleration $=0.5 \: m/sec^3$.

Assume total reaction time $=2.5$ seconds; coefficient of longitudinal friction of the pavement $=0.35$; height of head light of the vehicle $=0.75$ m; and beam angle $1^{\circ}$.

What is the length of valley curve (in m) based on the head light sight distance condition? ___________