
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 $f(x)=\frac{1}{2}-\frac{1}{2x}$ and $g(x)=a\ln x$, where $a>0$.
$(1)$ If $a=\frac{1}{2}$, and $f(x) < g(x)$ holds for all $x$ in $(0,t)$, find the range of real number $t$.
$(2)$ If the graphs of functions $f(x)$ and $g(x)$ have exactly one common tangent line, find the value of $a$.

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