
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
After conducting a market survey, a tourist city found that in the past month (counted as 30 days), the number of tourists on the $t$-th day $(1 \leqslant t \leqslant 30, t \in \mathbb{N}^{*})$ approximately satisfies $f(t) = 4 + \frac{1}{t}$ (in ten thousand people), and the average spending per tourist $g(t)$ (in yuan) approximately satisfies $g(t) = 120 - |t - 20|$.

$(1)$ Find the function relationship between the city's daily tourism revenue $W(t)$ (in ten thousand yuan) and time $t (1 \leqslant t \leqslant 30, t \in \mathbb{N}^{*})$;

$(2)$ Find the minimum value of the city's daily tourism revenue.

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