
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 vectors $\overrightarrow{\alpha}$, $\overrightarrow{\beta}$, $\overrightarrow{\gamma}$ satisfy $|\overrightarrow{\alpha}|=1$, $|\overrightarrow{\alpha}-\overrightarrow{\beta}|=|\overrightarrow{\beta}|$, $(\overrightarrow{\alpha}-\overrightarrow{\gamma}) \cdot (\overrightarrow{\beta}-\overrightarrow{\gamma})=0$. If for every determined $\overrightarrow{\beta}$, the maximum and minimum values of $|\overrightarrow{\gamma}|$ are $m$ and $n$ respectively, then for any $\overrightarrow{\beta}$, the minimum value of $m-n$ is \_\_\_\_\_\_\_\_.

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