
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 vector $\vec{a} =(2,2)$, the angle between vector $\vec{b}$ and vector $\vec{a}$ is $\frac{3\pi}{4}$, and $\vec{a} \cdot \vec{b} =-2$,

$(1)$ Find vector $\vec{b}$;

$(2)$ If $\vec{t}=(1,0)$ and $\vec{b} \perp \vec{t}$, $\vec{c}=\left(\cos A,\cos^2 \frac{C}{2}\right)$, where $A$, $B$, $C$ are the internal angles of $\triangle ABC$, and if the three internal angles $A$, $B$, $C$ form an arithmetic sequence in order, try to find the range of $|\vec{b}+ \vec{c}|$.

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