
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
If the function $f(x)=\begin{cases} \left( \frac{1}{2} \right)^{x- \frac{1}{2}} & x\leqslant \frac{1}{2} \\ \log_{a}x & x > \frac{1}{2} \end{cases}$ ($a > 0$, and $a \neq 1$) has a range of $\mathbb{R}$, then the range of the real number $a$ is ______.

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