
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 proposition $p$: The equation $x^{2}-mx+1=0$ has two distinct positive real roots; proposition $q$: The equation $4x^{2}+4(m-2)x+m^{2}=0$ has no real roots; If "p or q" is true, and "p and q" is false, then the correct conclusions among the following are:  
$(1)p$ and $q$ are both true;  
$(2)p$ and $q$ are both false;  
$(3)p$ and $q$ are true and false respectively;  
$(4)At least one of p and q is true;$  
$(5)At least one of p and q is false.$  
The correct conclusion numbers are \_\_\_\_\_\_, and the range of $m$ is \_\_\_\_\_\_.

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