
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
In response to the national policy of expanding domestic demand, a manufacturer plans to hold a promotional event for a certain product in 2018. After investigation and calculation, the annual sales volume of the product (i.e., the annual production of the factory) $x$ (in ten thousand units) and the annual promotional expenses $t$ (in ten thousand yuan, $t \geqslant 0$) satisfy $x=4-\frac{k}{2t+1}$ (where $k$ is a constant). If no promotional activities are conducted, the annual sales volume of the product can only be 1 ten thousand units. It is known that the fixed investment for producing the product in 2018 is 6 ten thousand yuan, and an additional investment of 12 ten thousand yuan is required for every ten thousand units produced. The factory sets the selling price of each unit to 1.5 times the average production investment cost (including both the fixed and additional production investments).

(1) Find the constant $k$, and express the profit $y$ (in ten thousand yuan) of the factory for the product in 2017 as a function of the annual promotional expenses $t$ (in ten thousand yuan);

(2) How much should the factory invest in annual promotional expenses in 2018 to maximize its profit?

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