
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) \( f(m + f(f(n))) = -f(f(m+1) - n) \) for all integers \( m \) and \( n \);

(b) \( g \) is a polynomial function with integer coefficients and \( g(n) = g(f(n)) \) for all integers \( n \).

Determine \( f(1991) \) and the most general form of \( g \).

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