
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
For which values of \( p \) do the quadratic equations \( x^{2} - (p+1)x + (p+1) = 0 \) and \( 2x^{2} + (p-2)x - p - 7 = 0 \) have a common root? Find this root.

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