
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  $n$  points  $A_1, A_2, \ldots, A_n$ , ( $n>2$ ), be considered in the space, where no four points are coplanar. Each pair of points  $A_i, A_j$  are connected by an edge. Find the maximal value of  $n$  for which we can paint all edges by two colors – blue and red  such that the following three conditions hold:

**I.** Each edge is painted by exactly one color.
**II.** For  $i = 1, 2, \ldots, n$ , the number of blue edges with one end  $A_i$  does not exceed 4.
**III.** For every red edge  $A_iA_j$ , we can find at least one point  $A_k$  ( $k \neq i, j$ ) such that the edges  $A_iA_k$  and  $A_jA_k$  are blue.

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