
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A store sells $1$ unit of computer type $A$ and $2$ units of computer type $B$ for a profit of $400$ yuan. It sells $2$ units of computer type $A$ and $1$ unit of computer type $B$ for a profit of $350$ yuan. The store plans to purchase a total of $100$ computers of both types, with the quantity of type $B$ computers not exceeding twice the quantity of type $A$ computers. Let $x$ be the quantity of type $A$ computers purchased. The total profit $y$ in yuan from selling these $100$ computers can be expressed as a function of $x$. Find the function and determine the quantities of type $A$ and type $B$ computers that will maximize the total profit.

When the manufacturer reduces the factory price of type $A$ computers by $m$ yuan $(0 < m \leq 50)$ and limits the store to purchasing a maximum of $70$ type $A$ computers, design a purchasing plan that maximizes the total profit from selling these $100$ computers while keeping the selling price of each type of computer unchanged.

Present the answer in LaTex format: \boxed{Your answer}
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