
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 *Revolution* $(x) = x^3 +Ux^2 +Sx + A$ , where  $U$ ,  $S$ , and  $A$  are all integers and  $U +S + A +1 = 1773$ . Given that *Revolution* has exactly two distinct nonzero integer roots  $G$  and  $B$ , find the minimum value of  $|GB|$ .

*Proposed by Jacob Xu*

<details><summary>Solution</summary>*Solution.*  $\boxed{392}$ Notice that  $U + S + A + 1$  is just *Revolution* $(1)$  so *Revolution* $(1) = 1773$ . Since  $G$  and  $B$  are integer roots we write *Revolution* $(X) = (X-G)^2(X-B)$  without loss of generality. So Revolution $(1) = (1-G)^2(1-B) = 1773$ .  $1773$  can be factored as  $32 \cdot 197$ , so to minimize  $|GB|$  we set  $1-G = 3$  and  $1-B = 197$ . We get that  $G = -2$  and  $B = -196$  so  $|GB| = \boxed{392}$ .</details>

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