
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 points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) \) is \( \sqrt{30} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) + 3 \) is \( \sqrt{46} \). Find the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) + 1 \).

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