
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 function $f(x) = \begin{cases} \log_{a}x, & x<t \\ x^{2}-6x+10, & x\geq t \end{cases}$ ($a>0$, $a\neq 1$), if for all $t\in (2,3)$, there exists $y_0\in \mathbb{R}$, such that $f(x)=y_0$ has three distinct real roots, determine the range of the real number $a$.

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