
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 store sells a certain item at a cost price of $30$ yuan per item. The relevant information on the daily selling price and quantity sold of this item in the past $60$ days is shown in the table below:

|           | Time: $x$ (days) | Time: $x$ (days) |
|-----------|------------------|------------------|
| Selling Price (yuan per item) | $0.5x+35$ | $50$ |
| Quantity Sold (items) | $128-2x$ | $128-2x$ |

($1\leqslant x\leqslant 60$, $x$ is an integer) Let the daily profit of the item be $w$ yuan.
$(Ⅰ)$ Write the functional relationship between $w$ and $x$;
$(Ⅱ)$ On which day does the item have the maximum daily profit? What is the maximum daily profit?

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