
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 certain fast food restaurant offers two meal sets, $A$ and $B$. Meal set $A$ has a profit of $8$ yuan per serving and sells $90$ servings per day; meal set $B$ has a profit of $10$ yuan per serving and sells $70$ servings per day. If the price of meal set $A$ is increased by $1$ yuan per serving, $4$ fewer servings are sold per day; if the price of meal set $B$ is increased by $1$ yuan per serving, $2$ fewer servings are sold per day. (Note: the cost of the two meal sets remains the same) <br/> $(1)$ If the price of each meal set is increased by $x$ yuan, find the functional relationship between the total daily profits $w_{A}$ yuan, $w_{B}$ yuan of selling meal sets $A$ and $B$, and $x$; <br/> $(2)$ The management department stipulates that the sum of the price increases for these two meal sets is $10$ yuan. How much should the price of meal set $A$ be increased by for the sum of the daily profits of these two meal sets to be maximized?

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