
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 to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, 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 $f(A)$.
2. If the angle between $\vec{m}$ and $\vec{n}$ is $\frac{\pi}{3}$, $C=\frac{\pi}{3}$, and $c=\sqrt{6}$, find the value of $b$.

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