
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
Given a linear function \( f(x) \), it is known that the distance between the intersection points of the graphs \( y=x^{2}+2 \) and \( y=f(x) \) is \(\sqrt{10}\), and the distance between the intersection points of the graphs \( y=x^{2}-1 \) and \( y=f(x)+1 \) is \(\sqrt{42}\). Find the distance between the intersection points of the graphs \( y=x^{2} \) and \( y=f(x)+1\).

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