
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 $f(x) = 2\cos^2x + 2\sqrt{3}\sin x\cos x$, $g(x) = xe^{-x}$.
(1) When $x \in \mathbb{R}$, find the intervals where the function $f(x)$ is monotonically decreasing;
(2) If for any $x_1 \in [1, 3]$, $x_2 \in [0, \frac{\pi}{2}]$, the inequality $g(x_1) + a + 3 > f(x_2)$ always holds, find the range of the real number $a$.

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