
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
We call a graph with n vertices  $k-flowing-chromatic$  if:
1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors.
2. we can choose a hamilton cycle  $v_1,v_2,\cdots , v_n$ , and move the chess on  $v_i$  to  $v_{i+1}$  with  $i=1,2,\cdots ,n$  and  $v_{n+1}=v_1$ , such that any two neighboring chess also have different colors.
3. after some action of step 2 we can make all the chess reach each of the n vertices.
Let T(G) denote the least number k such that G is k-flowing-chromatic.
If such k does not exist, denote T(G)=0.
denote  $\chi (G)$  the chromatic number of G.
Find all the positive number m such that there is a graph G with  $\chi (G)\le m$  and  $T(G)\ge 2^m$  without a cycle of length small than 2017.

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