
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
In a certain game, Auntie Hall has four boxes  $B_1$ ,  $B_2$ ,  $B_3$ ,  $B_4$ , exactly one of which contains a valuable gemstone; the other three contain cups of yogurt. You are told the probability the gemstone lies in box  $B_n$  is  $\frac{n}{10}$  for  $n=1,2,3,4$ . 

Initially you may select any of the four boxes; Auntie Hall then opens one of the other three boxes at random (which may contain the gemstone) and reveals its contents. Afterwards, you may change your selection to any of the four boxes, and you win if and only if your final selection contains the gemstone. Let the probability of winning assuming optimal play be  $\tfrac mn$ , where  $m$  and  $n$  are relatively prime integers.  Compute  $100m+n$ .

*Proposed by Evan Chen*

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