
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$  and  $ g$  be two integer-valued functions defined on the set of all integers such that

*(a)*  $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$  for all integers  $ m$  and  $ n;$ 
*(b)*  $ g$  is a polynomial function with integer coefficients and g(n) =  $ g(f(n))$   $ \forall n \in \mathbb{Z}.$ 

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