
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  $a, b, c$  be three distinct positive integers. Define  $S(a, b, c)$  as the set of all rational roots of  $px^2 + qx + r = 0$  for every permutation  $(p, q, r)$  of  $(a, b, c)$ . For example,  $S(1, 2, 3) = \{ -1, -2, -1/2 \}$  because the equation  $x^2+3x+2$  has roots  $-1$  and  $-2$ , the equation  $2x^2+3x+1=0$  has roots  $-1$  and  $-1/2$ , and for all the other permutations of  $(1, 2, 3)$ , the quadratic equations formed don't have any rational roots.

Determine the maximum number of elements in  $S(a, b, c)$ .

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