
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
A line  $g$  is given in a plane.  $n$  distinct points are chosen arbitrarily from  $g$  and are named as  $A_1, A_2, \ldots, A_n$ . For each pair of points  $A_i,A_j$ , a semicircle is drawn with  $A_i$  and  $A_j$  as its endpoints. All semicircles lie on the same side of  $g$ . Determine the maximum number of points (which are not lying in  $g$ ) of intersection of semicircles as a function of  $n$ .

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