
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 following table shows the daily sales price and sales volume information of this item over a period of $60$ days:

| | Time: $x$ (days) | |
|---|---|---|
| | $1\leqslant x\leqslant 30$ | $31\leqslant x\leqslant 60$ |
| Daily Sales Price (yuan per item) | $0.5x+35$ | $50$ |
| Daily Sales Volume (items) | $124-2x$ | $124-2x$ |

$(1\leqslant x\leqslant 60$, where $x$ is an integer)  
Let the daily profit of this item be $w$ yuan.  
$(1)$ Write down the function relationship between $w$ and $x$ directly.  
$(2)$ 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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