
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 venture capital firm plans to invest in a new energy product, estimating the range of investment returns to be $[10,100]$ (unit: ten thousand yuan). They are now preparing a reward scheme for the research team: the reward amount $y$ (unit: ten thousand yuan) increases with the increase in investment returns $x$ (unit: ten thousand yuan) and the reward amount does not exceed 5 ten thousand yuan, while also not exceeding 20% of the investment returns.

$(1)$ If a function model $y=f(x)$ is established to formulate the reward scheme, please write down the conditions that the reward function model should meet based on the problem statement;

$(2)$ There are two reward function models: $(i) y= \frac{1}{20}x+1$;

$(ii) y=\log_{2}x-2.$ Try to analyze whether these two function models meet the company's requirements.

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