
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
Julia possesses a deck of 18 cards, numbered from 1 to 18. After shuffling, she distributes the cards face down in 3 rows and 6 columns.

Julia chooses a sum \(X\) and flips two cards. If the sum is \(X\), she removes the pair from the table, but if it is not \(X\), she returns the cards to their original positions. She repeats this process until all pairs with a sum equal to \(X\) are flipped.

a) If she chooses \(X=8\), which pairs will be removed from the table?

b) For which value of \(X\) will all the cards be removed from the table?

c) For a specific \(X\), exactly 2 cards remain on the table. How many possible values of \(X\) are there?

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