
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
Plane vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ satisfy: $|\vec{a}|=1, \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}=1$, and $|\vec{a}-\vec{b}+\vec{c}| \leq 2 \sqrt{2}$. Then the range of values for $\vec{a} \cdot \vec{c}$ is __.

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