
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
Let \\(f(x)=e^{x}\sin x\\) and \\(g(x)=(x+1)\cos x- \sqrt {2}e^{x}\\).
\\((1)\\) Find the intervals of monotonicity for \\(f(x)\\);
\\((2)\\) For all \\(x_{1} \in [0, \frac {\pi}{2}]\\) and \\(x_{2} \in [0, \frac {\pi}{2}]\\), such that \\(f(x_{1})+g(x_{2}) \geqslant m\\) holds, find the range of the real number \\(m\\);
\\((3)\\) Let \\(h(x)= \frac {2x}{\sin x} \cdot f(x)-n \cdot \sin 2x\\) have a unique zero in \\((0, \frac {\pi}{2})\\), find the range of the positive real number \\(n\\).

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