
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 a connected and locally finite graph \( G \), and a cut \( F \subseteq E(G) \) that divides the vertex set into \( V_1 \) and \( V_2 \). 

(i) If \( F \) is finite, then \( \overline{V_1} \cap \overline{V_2} = \emptyset \), and no arc in \( |G| \backslash \stackrel{\circ}{F} \) has one endpoint in \( V_1 \) and the other in \( V_2 \).

(ii) If \( F \) is infinite, then \( \overline{V_1} \cap \overline{V_2} \neq \emptyset \), and such an arc may exist.

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