
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 $\overrightarrow{a}=( \sqrt {2},m)(m > 0)$, $\overrightarrow{b}=(\sin x,\cos x)$, and the maximum value of the function $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$ is $2$.
1. Find $m$ and the smallest positive period of the function $f(x)$;
2. In $\triangle ABC$, $f(A- \frac {\pi}{4})+f(B- \frac {\pi}{4})=12 \sqrt {2}\sin A\sin B$, where $A$, $B$, $C$ are the angles opposite to sides $a$, $b$, $c$ respectively, and $C= \frac {\pi}{3}$, $c= \sqrt {6}$, find the area of $\triangle ABC$.

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