
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

After defeating Koschei, Ivan demanded gold to ransom Vasilisa from the bandits. Koschei took him to a cave and said: "There are gold bars in the chest. But you can't just take them: they are enchanted. Take one or several bars for yourself. Then I will put one or several bars back into the chest, but always a different number. We will take turns doing this: you put bars into your bag, I put them back into the chest, each time a different number. When it becomes impossible to move bars in this way anymore, you can take your bag of bars." What is the maximum number of bars Ivan can take, no matter what Koschei does, if the initial number of bars in the chest is a) 13; b) 14? How can he achieve this?

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