
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
Let $$f(x)= \frac {x^{2}-3x+8}{2} \quad (x≥2)$$, and $g(x)=a^{x} \quad (x>2)$.  
(1) If there exists $x_{0} \in [2, +\infty)$ such that $f(x_{0})=m$ holds, then the range of real number $m$ is \_\_\_\_\_\_  
(2) If for every $x_{1} \in [2, +\infty)$, there exists $x_{2} \in (2, +\infty)$ such that $f(x_{1})=g(x_{2})$, then the range of real number $a$ is \_\_\_\_\_\_.

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