
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 quadratic function $f(x) = ax^2 + bx + c$, define two new functions $u(x) = -f(-x)$, and $v(x) = f(x+1)$. Determine the number of intersection points between $y = f(x)$ and $y = u(x)$, and between $y = f(x)$ and $y = v(x)$. Let $m$ be the number of intersection points between the graphs of $y = f(x)$ and $y = u(x)$, and $n$ be the number of intersection points between the graphs of $y = f(x)$ and $y = v(x)$. Calculate $7m + 3n$.

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