
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 plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{e}$ satisfying $|\overrightarrow{e}|=1$, $\overrightarrow{a} \cdot \overrightarrow{e}=2$, $\overrightarrow{b} \cdot \overrightarrow{e}=3$, $|\overrightarrow{a}-\overrightarrow{b}|= \sqrt{5}$, then the minimum value of $\overrightarrow{a} \cdot \overrightarrow{b}$ is \_\_\_\_\_\_.

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