
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 rectangle \(ABCD\), side \(AB\) is three times longer than side \(BC\). Inside the rectangle, there is a point \(N\) such that \(AN = \sqrt{2}\), \(BN = 4\sqrt{2}\), and \(DN = 2\). Find the cosine of angle \(BAN\) and the area of rectangle \(ABCD\).

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