
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 $ABC$, with angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$ respectively, let $\vec{m} = (1,1)$ and $\vec{n} = (-\cos A, \sin A)$. Define $f(A) = \vec{m} \cdot \vec{n}$.
(1) Find the range of values for $f(A)$.
(2) If the angle between $\vec{m}$ and $\vec{n}$ is $\frac{\pi}{3}$, and $C = \frac{\pi}{3}$ with $c = \sqrt{6}$, find the value of $b$.

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