
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
Consider the function
\[ f(x) = \max \{-7x - 23, 2x + 5, 5x + 17 \} \]
defined for all real $x$. Let $q(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $x_4$, $x_5$, and $x_6$. Find $x_4 + x_5 + x_6$.

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