
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
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled  $A$ . The three vertices adjacent to vertex  $A$  are at heights 10, 11, and 12 above the plane. The distance from vertex  $A$  to the plane can be expressed as  $\tfrac{r-\sqrt{s}}{t}$ , where  $r$ ,  $s$ , and  $t$  are positive integers, and  $r+s+t<1000$ . Find  $r+s+t$ .

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