
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
Let \( p(x) \equiv x^3 + ax^2 + bx - 1 \) and \( q(x) \equiv x^3 + cx^2 + dx + 1 \) be polynomials with integer coefficients. Let \( \alpha \) be a root of \( p(x) = 0 \). The polynomial \( p(x) \) is irreducible over the rationals. Given that \( \alpha + 1 \) is a root of \( q(x) = 0 \), find an expression for another root of \( p(x) = 0 \) in terms of \( \alpha \), but not involving \( a \), \( b \), \( c \), or \( d \).

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