
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 that the domain of the function $f(x)$ is $D$, if for all $x_{1} \in D$, there exists $x_{2} \in D$ such that $\frac{x_{1}+f(x_{2})}{2}=a$, determine the range of real numbers for $m$ such that the function defined on $(0,+\infty)$, $f(x)=-x^{2}+mx-3$, has property $P(\frac{1}{2})$.

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