
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
The profits obtained from selling two types of commodities, A and B, are P (in ten thousand yuan) and Q (in ten thousand yuan), respectively. Their relationship with the invested capital t (in ten thousand yuan) is determined by the empirical formulas $P = \frac{1}{5}t$, $Q = \frac{3}{5}\sqrt{t}$. Now, 3 ten thousand yuan of capital is invested in operating commodities A and B, with x ten thousand yuan invested in commodity A.

1. Establish the function relation of the total profit y (in ten thousand yuan) with respect to x.
2. How much should be invested in commodity A, in terms of x (in ten thousand yuan), to achieve the maximum total profit y (in ten thousand yuan)?

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