
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 $x^2-2\sqrt{2}x+m=0$ has two distinct real roots; proposition $q$: $2^{m+1} < 4$  
$(1)$ If $p$ is true, find the range of real numbers $m$;  
$(2)$ If $p \vee q$ is true, and $p \wedge q$ is false, find the range of real numbers $m$.

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