Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: Consider the function
\[f(x) = \max \{-11x - 37, x - 1, 9x + 3\}\]defined for all real $x.$  Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $x_1,$ $x_2,$ $x_3.$  Find $x_1 + x_2 + x_3.$
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