
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 functions f(x) = xlnx and g(x) = ax^2 + 2ax.
1. Find the monotonic intervals of f(x).
2. If f'(x) is the derivative of the function f(x), and 2f'(x) ≤ g(x) - 2x holds true for all x in the domain, find the minimum value of the integer a.

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