
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
There are two types of products, A and B, with profits of $p$ ten-thousand yuan and $q$ ten-thousand yuan, respectively. Their relationship with the invested capital $x$ ten-thousand yuan is: $p= \frac{1}{5}x$, $q= \frac{3}{5} \sqrt{x}$. Now, with an investment of 3 ten-thousand yuan in managing these two products, how much capital should be allocated to each product in order to maximize profit, and what is the maximum profit?

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