
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 a linear function in terms of $x$: $y=(2m+1)x+m-3$.  
$(1)$ If the graph of the function passes through the origin, find the value of $m$.  
$(2)$ If the graph of the function is parallel to the line $y=3x-3$, find the value of $m$.  
$(3)$ For what value of $m$ does the graph of the function intersect the $y$-axis below the $x$-axis?

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