
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 magician and an assistant have a deck of cards. Each card has the same design on one side (the "back"), while the other side is colored in one of 2017 colors (there are 1,000,000 cards of each color in the deck). The magician and the assistant aim to perform the following trick. The magician leaves the room, and the audience places a row of \( n > 1 \) cards face down on the table. The assistant looks at these cards and then turns over all but one card, leaving them in the same order. The magician then enters, looks at the table, identifies one of the face-down cards, and guesses its color. What is the smallest number \( k \) that the magician can prearrange with the assistant so that the trick is guaranteed to succeed?

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