
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 grandmother and her beloved grandson Vasya agreed to show a trick to his mom. The grandmother had 10 different fillings for pastries, and she baked one pastry with each pair of these fillings. There were 45 pastries in total. By looking at a pastry, only the grandmother can determine its fillings.

The grandmother places $n$ out of the 45 pastries on a tray, and Vasya breaks them open and looks at their fillings. Then he tells his mom: "For any of the remaining pastries, I can definitely determine at least one of its fillings without breaking it open!" Indeed, the mom randomly picks any of the remaining pastries, and Vasya correctly determines one of its fillings.

What is the smallest $n$ for which Vasya and the grandmother can show such a trick to his mom? Answer: 36.

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