
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 $f(x) = x\ln x$.
$(1)$ Find the intervals of monotonicity and extreme values of the function $f(x)$;
$(2)$ If the tangent line $l$ passing through the point $M(x_0, f(x_0))(x_0 > \frac{1}{e})$ intersects the $x$-axis and $y$-axis at points $A$ and $B$, respectively, with the origin $O$, and the area of triangle $AOB$ is denoted as $S$, find the minimum value of $S$;
$(3)$ Let $g(x) = e^{\lambda x\ln x} - f(x)$, and the inequality $g(x) \geq 1$ holds for any $x \in (0, +\infty)$. Find the value of the real number $\lambda$.

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