
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) = ln(x) + 2x^2 + mx, g(x) = e^x + 3x^2 - nx, (m, n ∈ R).
(1) Discuss the number of extreme points of f(x);
(2) If ∀x > 0, the inequality f(x) ≤ g(x) always holds, when m, n are positive numbers, find the minimum value of $\frac{1}{m} + \frac{4}{n}$.

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