
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
In \\(\triangle ABC\\), let the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) be \\(a\\), \\(b\\), and \\(c\\) respectively. Let vector \\( \overrightarrow{m}=(\cos A+ \sqrt {2},\sin A)\\) and vector \\( \overrightarrow{n}=(-\sin A,\cos A)\\). If \\(| \overrightarrow{m}+ \overrightarrow{n}|=2\\),
\\((1)\\) find the magnitude of angle \\(A\\);
\\((2)\\) if \\(b=4 \sqrt {2}\\) and \\(c= \sqrt {2}a\\), find the area of \\(\triangle ABC\\).

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