
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
A bike-sharing company, "Mobike," plans to invest a total of 120 million yuan in two cities, A and B. According to industry regulations, each city must receive a minimum investment of 40 million yuan. Market research has shown that the return (P) on investment (a, in millions of yuan) in city A follows the equation P = $3\sqrt{2a} - 6$, and the return (Q) on investment in city B follows the equation Q = $\frac{1}{4}a + 2$. If we denote the investment in city A as x (in millions of yuan), the total return for the company is given by the function f(x) (in millions of yuan).

(1) Calculate the total return for the company when city A receives an investment of 50 million yuan.

(2) Determine the optimal investment strategy for cities A and B to maximize the total return.

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