
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 $a=(\cos x + \sin x, \sin x)$ and $b=(\cos x - \sin x, 2\cos x)$, let $f(x) = a \cdot b$.

1. Find the interval(s) where the function $f(x)$ is monotonically increasing.
2. In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $A=\frac{\pi}{3}$, $f(B) = 1$, and $\sqrt{3}a + \sqrt{2}b = 10$, find $c$.

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