Ms$.$ Anderson is the owner of a boutique bakery and wants to maximize her monthly profit by determining the number of cakes and pastries she should produce. Ms$.$ Anderson has a budget of $$10,000$ to spend on ingredients and labor costs. The cost of ingredients for a cake is $$20,$ and the cost for a pastry is $$5.$ The labor cost for a cake is $$15,$ and the labor cost for a pastry is $$3.$

Based on historical records, Ms$.$ Anderson estimates that she can sell at least $50$ cakes but no more than $85$ cakes in a month. She can sell at least $10$ pastries but no more than $20$ pastries in a month. Ms$.$ Anderson also knows that the demand for cakes is at least four times the demand for pastries.

What is the optimal number of cakes and pastries that Ms$.$ Anderson should produce to maximize her monthly profit, given the contribution for each cake and pastry are $$10$ and $$7\mathrm{.}5,$ respectively?

Note: Assume a month consists of $30$ days.