
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 the functions $f(x)=x^2-m\ln x$ and $h(x)=x^2-x+a$, and the function $f(x)$ has an extreme value at $x=1$.

$(1)$ Find the value of the real number $m$;

$(2)$ Let the function $k(x)=f(x)-h(x)$. If the function $k(x)$ has exactly two different zeros in the interval $[1,3]$, find the range of values for the real number $a$.

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