
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
I1.1 Let \( P \) be the unit digit of \( 3^{2003} \times 5^{2002} \times 7^{2001} \). Find the value of \( P \).

I1.2 If the equation \( \left(x^{2}-x-1\right)^{x+P-1}=1 \) has \( Q \) integral solutions, find the value of \( Q \).

I1.3 Let \( x, y \) be real numbers and \( x y=1 \).
If the minimum value of \( \frac{1}{x^{4}}+\frac{1}{Q y^{4}} \) is \( R \), find the value of \( R \).

11.4 Let \( x_R, x_{R+1}, \ldots, x_K (K>R) \) be \( K-R+1 \) distinct positive integers and \( x_R + x_{R+1} + \ldots + x_K = 2003 \). If \( S \) is the maximum possible value of \( K \), find the value of \( S \).

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