
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
Let  $M$  be a set consisting of  $n$  points in the plane, satisfying:
i) there exist  $7$  points in  $M$  which constitute the vertices of a convex heptagon; 
ii) if for any  $5$  points in  $M$  which constitute the vertices of a convex pentagon, then there is a point in  $M$  which lies in the interior of the pentagon.
Find the minimum value of  $n$ .

*Leng Gangsong*

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