
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
Due to the recent impact of a typhoon, a certain city has incurred significant economic losses. To recuperate, a manufacturer plans to hold a large-scale promotional event. It has been calculated that, when the promotional expenses for a certain product are x ten-thousand yuan, the sales volume t (in ten-thousand units) satisfies $t = 5 - \frac{2}{x+1}$, where $0 \leq x \leq a^2 - 3a + 3$ with $a > 0$. Assuming that production volume is equal to the sales volume, it is known that producing t ten-thousand units of the product incurs a cost of $(10 + 2t)$ ten-thousand yuan (not including promotional expenses), and the selling price of the product is set at $(4 + \frac{20}{t})$ ten-thousand yuan per ten-thousand units.
(1) Express the profit y (in ten-thousand yuan) as a function of the promotional expenses x (in ten-thousand yuan).
(2) How much should be spent on promotional expenses to maximize the manufacturer's profit?

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