
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
Given two parallel lines $4x-2y+7=0$ and $2x-y+1=0$, the distance between them is equal to half the distance from the origin $O$ to the line $l$: $x-2y+m=0$ ($m > 0$).  
$(1)$ Find the value of $m$;  
$(2)$ Determine the positional relationship between line $l$ and circle $C$: $x^{2}+(y-2)^{2}= \frac{1}{5}$.

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