
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 issue of food safety is increasingly attracting people's attention. The overuse of pesticides and fertilizers poses certain health risks to the public. To provide consumers with reassuring vegetables, a certain rural cooperative invests 2 million yuan annually to build two pollution-free vegetable greenhouses, A and B. Each greenhouse requires at least an investment of 200,000 yuan. Greenhouse A grows tomatoes, and greenhouse B grows cucumbers. According to past vegetable planting experience, the annual income from growing tomatoes, $P$, and the annual income from growing cucumbers, $Q$, satisfy the relationships $P = 80 + 4\sqrt{2a}$ and $Q = \frac{1}{4}a + 120$, respectively, where $a$ represents the investment in units of 10,000 yuan. If the investment in greenhouse A is represented by $x$ (in units of 10,000 yuan), the total annual income from both greenhouses is represented by $f(x)$.

(1) Determine the value of $f(50)$;
(2) Determine how to allocate the investments for greenhouses A and B to maximize the total income $f(x)$.

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