
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
 $N $  different numbers are written on blackboard and one of these numbers is equal to  $0$ .One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between  $-2016$  and  $2016$  were written on blackboard(and some other numbers maybe). Find  the smallest possible value of  $N $ .

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