
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 functions $f(x)=x+m$ and $g(x)=x^2-mx+\frac{m^2}{2}+2m-3$.

$(1)$ If the solution set of $g(x)<\frac{m^2}{2}+1$ is $(1,a)$, find the value of $a$.

$(2)$ If for all $x_{1}\in [0,1]$, there exists $x_{2}\in [1,2]$ such that $f(x_{1}) > g(x_{2})$, find the range of real numbers for $m$.

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