
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
According to market research, the relationship between the selling price (in yuan) of a certain product in a supermarket and the time $t$ (in days) in the past month (30 days) is approximately satisfied by the function $f(t) = 100(1+\frac{1}{t})$, and the relationship between the sales volume (in units) and the time $t$ (in days) is approximately satisfied by the function $g(t) = 125-|t-25|$.
$(1)$ Write down the expression of the daily sales amount $w(t)$ of the product as a function of time $t$ $(1\leq t\leq 30, t\in \mathbb{N})$;
$(2)$ Find the maximum and minimum values of the daily sales amount $w(t)$ of the product.

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