A model used to estimate the time it will take to hike to the top of a mountain on a trail is of the form $T=aL+bG$, where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $50$ minutes to hike to the top if a trail is $2$ miles long and ascends $500$ feet. It also estimates that it will take $54$ minutes if a trail is $2.5$ miles long and ascends $200$ feet. How many minutes does the model estimate it will take to hike to the top if the trail is $5$ miles long and ascends $1000$ feet?