
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
**(i)** Determine the smallest number of edges which a graph of  $ n$  nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges).
**(ii)** Determine the smallest number of edges which a graph of  $ n$  nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).

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