
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
Given that the profits obtained from selling "laptops" and "desktop computers" are P (unit: ten thousand yuan) and Q (unit: ten thousand yuan) respectively, their relationship with the purchase capital t (unit: ten thousand yuan) follows the empirical formulas $P= \frac{1}{16}t$ and $Q= \frac{1}{2}t$. A mall decides to invest a purchase capital of 500 thousand yuan, entirely for buying these two types of computers. How should the mall allocate the purchase capital to maximize the profit y (unit: ten thousand yuan) from selling the computers? What is the maximum profit in ten thousand yuan?

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