
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
For a graph  $G$  on  $n$  vertices, let  $P_G(x)$  be the unique polynomial of degree at most  $n$  such that for each  $i=0,1,2,\dots,n$ ,  $P_G (i)$  equals the number of ways to color the vertices of the graph  $G$  with  $i$  distinct colors such that no two vertices connected by an edge have the same color. For each integer  $3\le k \le 2017$ , define a  $k$ -*tasty* graph to be a connected graph on  $2017$  vertices with  $2017$  edges and a cycle of length  $k$ . Let the *tastiness* of a  $k$ -tasty graph  $G$  be the number of coefficients in  $P_G(x)$  that are odd integers, and let  $t$  be the minimal tastiness over all  $k$ -tasty graphs with  $3\le k \le 2017$ . Determine the sum of all integers  $b$  between  $3$  and  $2017$  inclusive for which there exists a  $b$ -tasty graph with tastiness  $t$ .

*Proposed by Vincent Huang*

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