
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 $y=\frac{1}{2}x^{2}$, the tangent line at the point $\left( x_{0},\frac{1}{2}x_{0}^{2} \right)$ is $l$. If line $l$ is also a tangent line to the graph of the function $y=\ln x$ $(0 < x < 1)$, determine the range of values for $x_{0}$.

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