
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 $y=x^2$, its graph has a tangent line $l$ at the point $(x_0, x_0^2)$. If $l$ is also tangent to the graph of the function $y=\ln x, x\in (0,1)$, determine the value of the inequality $0 < x_0 < \frac{1}{2}$.

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