
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
**i.)** Let  $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$  What is the remainder when the polynomial  $g(x^{12}$  is divided by the polynomial  $g(x)$ ?

**ii.)** If  $k$  is a positive number and  $f$  is a function such that, for every positive number   $x, f(x^2 + 1 )^{\sqrt{x}} = k.$  Find the value of 

\[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number  $y.$ 

**iii.)** The function  $f$  satisfies the functional equation  $f(x) + f(y) = f(x+y) - x \cdot y - 1$  for every pair  $x,y$  of real numbers. If  $f(1) = 1,$  then find the numbers of integers  $n,$  for which  $f(n) = n.$ 

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