
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
Determine the polynomial  $$ f(x) = x^k + a_{k-1} x^{k-1}+\cdots +a_1 x +a_0  $$  of smallest degree such that  $a_i \in \{-1,0,1\}$  for  $0\leq i \leq k-1$  and  $f(n)$  is divisible by  $30$  for all positive integers  $n$ .

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