
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 positive integers with  $a \le 10$ .  Suppose the parabola  $y = ax^2 + bx + c$  meets the  $x$ -axis at two distinct points  $A$  and  $B$ .  Given that the length of  $\overline{AB}$  is irrational, determine, with proof, the smallest possible value of this length, across all such choices of  $(a, b, c)$ .

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