
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  $ f ( x ) \in \mathbb { Z } [ x ] $  be a polynomial with integer coefficients such that  $ f ( 1 ) = - 1 , f ( 4 ) = 2 $  and  $f ( 8 ) = 34 $ . Suppose  $n\in\mathbb{Z}$  is an integer such that  $ f ( n ) = n ^ { 2 } - 4 n - 18 $ . Determine all possible values for  $n$ .

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