
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
On a table, there are 30 coins: 23 ten-ruble coins and 7 five-ruble coins. Out of these, 20 coins are heads up, and the remaining 10 are tails up. What is the smallest number \( k \) such that among any randomly chosen \( k \) coins, there will always be at least one ten-ruble coin that is heads up?

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