
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 graph of the function $y=x^2$ at the point $(x_0, x_0^2)$, where $x_0 \in (\sqrt{n}, \sqrt{n+1})$ and $n \in \mathbb{N}$, the tangent line at this point is denoted as $l$. If $l$ is also tangent to the graph of the function $y=\ln x$ for $x \in (0,1)$, then find the value of $n$.

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