
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

A linear function \( f(x) \) is given. It is known that the distance between the points of intersection of the graphs \( y = x^{2} \) and \( y = f(x) \) is \( 2 \sqrt{3} \), and the distance between the points of intersection of the graphs \( y = x^{2}-2 \) and \( y = f(x)+1 \) is \( \sqrt{60} \). 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