
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 that the plane vectors $\overrightarrow{\mathrm{a}}$, $\overrightarrow{\mathrm{b}}$, and $\overrightarrow{\mathrm{c}}$ satisfy $|\overrightarrow{\mathrm{a}}|=\frac{|\overrightarrow{\mathrm{b}}|}{2}=|\overrightarrow{\mathrm{c}}|=1$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=1$, find the range of the expression $\left|\overrightarrow{\mathrm{c}}+\frac{\vec{a}}{2}\right| + \frac{1}{2}|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{b}}|$.

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