
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 $f(x)=2x\ln x$.
(1) Find the monotonic intervals of the function $f(x)$;
(2) Find the equation of the tangent line to the graph of the function $f(x)$ that passes through the point $(0,-2)$;
(3) Find the range of the constant $\lambda$ such that $f(x) < \lambda(x^{2}-1)$ holds for all $x\in(1,+\infty)$.

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