
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
\( p(x, y, z) \) is a polynomial with real coefficients such that:
1. \( p(tx, ty, tz) = t^2 f(y - x, z - x) \) for all real \( x, y, z, t \) (and some function \( f \));
2. \( p(1, 0, 0) = 4 \), \( p(0, 1, 0) = 5 \), and \( p(0, 0, 1) = 6 \);
3. \( p(\alpha, \beta, \gamma) = 0 \) for some complex numbers \( \alpha, \beta, \gamma \) such that \( |\beta - \alpha| = 10 \).

Find \( |\gamma - \alpha| \).

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