
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
In 2022, the global epidemic raged, medical supplies were in short supply, frontline protesters risked their lives, using their own bodies to open a safe path for us. Until November, various regions of the country successively announced the lifting of the lockdown, and various industries resumed work and production. "Sunshine Medical Equipment Factory" immediately started production. The table below shows the relationship between the selling price $y$ (in yuan per set) of protective suits and the sales volume $t$ (sets) for the first 5 days of December:

| Day $x$ | 1 | 2 | 3 | 4 | 5 |
|---------|---|---|---|---|---|
| Selling Price $y$ (yuan/set) | 30 | 32 | 34 | 36 | 38 |
| Sales Volume $t$ (sets) | 100 | 120 | 140 | 160 | 180 |

Due to the chaos discovered by the pricing department, starting from the 6th day, the factory adjusted the price to 28 yuan per set. According to statistics, the relationship between the sales volume $t$ of protective suits after the 6th day and the day $x$ is given by $t = -x^2 + 50x - 100$ for $6 \leq x \leq 20$, where $x$ is an integer.

$(1)$ Write down the relationship between the sales volume $t$ and the day $x$ (for the first 4 days) directly. Also, find on which day after the 6th day the sales volume is maximized, and the maximum value is ______.

$(2)$ If the cost price is 22 yuan, write down the function relationship $W$ (in yuan) of the profit obtained by selling protective suits for these days (considering 20 days) with respect to $x$. Directly write down the day on which the profit is maximized.

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