
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 Hamlet performance, several roles can be doubled, for example, both Gertrude and the Player Queen can be played by two different actors. Before the performance, a draw determines which actor will play the Player Queen and Gertrude that day. Similar draws decide the actors for the other pairs of roles. Sári has already seen the performance once, but she wants to see the Gertrude/Player Queen, Claudius/Player King, and Ophelia/Fortinbras roles with the alternative casting, although not necessarily in a single performance. How many more performances should she attend to have at least a $90\%$ probability of seeing all three roles in the alternative casting as well?

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