
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 maximum breeding capacity of a fishery is $m$ tons. To ensure the growth space for the fish, the actual breeding amount $x$ must be less than $m$, leaving an appropriate idle capacity. The ratio of the idle capacity to the maximum breeding capacity is called the idle rate. It is known that the annual increase of the fish population $y$ (in tons) and the actual breeding amount $x$ (in tons) are directly proportional to the product of the idle rate (with the proportionality constant $k > 0$).  
$(1)$ Write the functional relationship between $y$ and $x$, and specify its domain;  
$(2)$ Find the maximum annual growth of the fish population;  
$(3)$ When the annual growth of the fish population reaches its maximum, determine the range of values for $k$.

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