
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 $\triangle ABC$, the sides opposite angles A, B, and C are denoted by $a$, $b$, $c$ respectively, and $\cos C= \frac{1}{5}$.

(I) Find the value of $\sin(2C+ \frac{\pi}{4})$;
(II) Given that $\overrightarrow{CA} \cdot \overrightarrow{CB} = 1$ and $a + b = \sqrt{37}$, find the length of side $c$ and the area of $\triangle ABC$.

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