
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
Two children take turns writing a $k$-digit number: the first digit is written by the first child, the second digit by the second child, the third digit again by the first child, and so on. Can the second child ensure that the resulting number is divisible by 9, even if the first child tries to interfere? Discuss the following two cases:

1. \( k = 10 \)
2. \( k = 13 \)

(Kyiv Mathematical Olympiad, 1975)

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