
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
Food safety issues are increasingly attracting people's attention. The abuse of pesticides and chemical fertilizers poses certain health risks to the public. To provide consumers with safe vegetables, a rural cooperative invests 2 million yuan each year to build two pollution-free vegetable greenhouses, A and B. Each greenhouse requires an investment of at least 200,000 yuan. Greenhouse A grows tomatoes, and Greenhouse B grows cucumbers. Based on past gardening experience, it has been found that the annual income $P$ from growing tomatoes and the annual income $Q$ from growing cucumbers with an investment of $a$ (unit: 10,000 yuan) satisfy $P=80+4\sqrt{2a}, Q=\frac{1}{4}a+120$. Let the investment in Greenhouse A be $x$ (unit: 10,000 yuan), and the total annual income from the two greenhouses be $f(x)$ (unit: 10,000 yuan).

  $(I)$ Calculate the value of $f(50)$;

  $(II)$ How should the investments in Greenhouses A and B be arranged to maximize the total income $f(x)$?

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