
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
Consider the graph on  $1000$  vertices  $v_1, v_2, ...v_{1000}$  such that for all  $1 \le i < j  \le 1000$ ,  $v_i$  is connected to  $v_j$  if and only if  $i$  divides  $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.

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