You are given the role of an investor with a budget of $$850.$ Your task is to develop an investment

strategy for the next $6$ months that maximizes your potential returns from investing in the stock

market and a savings account. You have identified three different company's stocks $-$ Acebook,

Bicrosoft, and Coogle $-$ each costing $$11$ per share. Additionally, you have the option to allocate a

portion of your budget to a savings account that offers a $2\mathrm{.}5\%$ semiannual interest rate.

There are two market scenarios you must consider for your investment in the next six months: a

bullish market and a bearish market. In the bullish market, Acebook stock is expected to yield a

$20\%$ return, Bicrosoft a $30\%$ return, and Coogle a $5\%$ return. In contrast, in the bearish market,

Acebook is expected to have a $-10\%$ return, Bicrosoft a $-5\%$ return, and Coogle a whopping $10\%$

return. In addition, in a bullish market, there is a sell limit of $20$ shares for Acebook, $30$ for

Bicrosoft, and $70$ for Coogle. In a bearish market, they all have a sell limit of $50$ shares.

Your task is to determine how many shares to purchase from each stock and how much money to

allocate to the savings account. The total cost of your investments should not exceed your budget

of $$850.$ Additionally, you cannot purchase a negative number of shares or allocate a negative

amount to the savings account.

Formulate this investment decision as a linear programming problem. Define your variables clearly,

set up the objective function to maximize your returns, and write down all the necessary constraints.

Remember, your goal is to devise a strategy that optimizes returns under both bullish and bearish

market conditions. Consider the potential returns from stock appreciation and interest earnings

from the savings account in each scenario. Your solution should reflect your assessment of the

market's potential behavior $($bullish or bearish$)$ over the next $6$ months and how best to allocate

your limited resources.