
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
We can apply the following transformations to the triplet $(A, B, C)$: (i) Permute the triplet arbitrarily; (ii) Change $(A, B, C)$ to $(2B + 2C - A, B, C)$. The initial state of the triplet is $(-1, 0, 1)$.

(1) Can we obtain the triplet $(2008, 2009, 2010)$ or $(2009, 2010, 2011)$ through a finite number of transformations? Explain your reasoning.

(2) Given that we can obtain the triplet $(1, 2024, x)$ through a finite number of transformations, find all possible values of $x$ and explain your reasoning.

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