
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]


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A certain factory produces a type of product at a rate of at least one ton per day. The products produced that day are all sold out on the same day, and the selling price changes with the production volume. When the daily production volume $x$ (with $x \geq 1$) is between 1 and 20 tons, the daily sales $y$ (in ten-thousand yuan) and the production volume $x$ satisfy the relation $y = a\ln x + b$. If the production volume exceeds 20 tons for the day, the sales can only be maintained at the level corresponding to a daily production of 20 tons. It is known that when the daily production is 2 tons the sales are 4.5 ten-thousand yuan, and when the production is 4 tons, the sales are 8 ten-thousand yuan.  
(1) Express the daily sales $y$ as a function of the daily production volume $x$.    
(2) If the daily production cost is $c(x) = \frac{1}{2}x + 1$ (in ten-thousand yuan), for what daily production volume can the daily profit reach a maximum? Also, find out the maximum value. (Note: When calculating, take $\ln 2 = 0.7$, $\ln 5 = 1.6$)

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