
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 $x^{2}-mx+m^{2}-2m+1=0$ has real roots with respect to $x$, and proposition $q: 1-2a \lt m \lt a+1$.  
$(1)$ If proposition $p$ is true, find the range of real number $m$;  
$(2)$ If $q$ is a sufficient condition for $p$, find the range of real number $a$.

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