
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
Say an integer polynomial is  $\textit{primitive}$  if the greatest common divisor of its coefficients is  $1$ .  For example,  $2x^2+3x+6$  is primitive because  $\gcd(2,3,6)=1$ .  Let  $f(x)=a_2x^2+a_1x+a_0$  and  $g(x) = b_2x^2+b_1x+b_0$ , with  $a_i,b_i\in\{1,2,3,4,5\}$  for  $i=0,1,2$ .  If  $N$  is the number of pairs of polynomials  $(f(x),g(x))$  such that  $h(x) = f(x)g(x)$  is primitive, find the last three digits of  $N$ .

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