
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 dosage of children's medication is often calculated based on their weight. For a certain medication, a child weighing $10kg$ should take $110mg$ per dose; a child weighing $15kg$ should take $160mg$ per dose. When the weight of a child is between $5$ and $50kg$, the dosage $y (mg)$ per dose is a linear function of the child's weight $x (kg)$. In reality, the actual dosage of the medication can be slightly higher than the normal dosage, but it cannot exceed $1.2$ times the normal dosage, otherwise it will cause significant harm to the child's body.  
$(1)$ Find the functional relationship between $y$ and $x$, and write down the range of values for the independent variable $x$;  
$(2)$ If one package size of the medication is $300mg/$ bag, determine the range of weights for children who can take one bag of medication at a time when they are sick.

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