
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
An international cosmetics manufacturing company plans to carry out a series of promotional activities during the 2016 Rio Olympics to capture a larger market share. Based on market survey and calculations, the annual sales volume $x$ (in 10,000 units) is inversely proportional to the annual promotional expenses $t$ (in 10,000 Yuan), with a proportionality of $3-x$ to $t+1$. If no promotional activities are conducted, the annual sales can only reach 1 (10,000 units). Considering that the fixed costs, including depreciation of equipment and maintenance, are 30,000 Yuan, and the production cost for every 10,000 units is 320,000 Yuan, if the selling price per unit is set as the sum of 150% of the production cost per unit and half of the average promotional cost per unit, then all produced cosmetics will be sold out for that year.

(1) Express the 2016 profit $y$ (in 10,000 Yuan) as a function of the promotional expenses $t$ (in 10,000 Yuan).
(2) How much should the company invest in promotional expenses in 2016 to maximize the annual profit?
(Note: Profit = Sales Revenue - Production Cost - Promotional Expenses, and Production Cost = Fixed Costs + Production Costs)

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