
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 the function $f(x) = m\sin x + \sqrt{2}\cos x$, ($m > 0$) has a maximum value of 2.  
(Ⅰ) Find the range of the function $f(x)$ on $[-2, -1] \cup (2, 6)$.  
(Ⅱ) Given that the radius $R$ of the circumcircle of $\triangle ABC$ is $\sqrt{3}$, and $f(A- \frac{\pi}{4}) + f(B- \frac{\pi}{4}) = 4\sqrt{6}\sin A \sin B$, where the sides opposite angles $A$ and $B$ are $a$ and $b$ respectively, find the value of $\frac{1}{a} + \frac{1}{b}$.

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