
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
Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations:
\begin{align*}
  11x +  by +  cz &= 0 \\
   ax + 24y +  cz &= 0 \\
   ax +  by + 41z &= 0.
\end{align*}
Suppose $ a \ne 11$ and $ x \ne 0$. What is the value of
\[ \frac{a}{a - 11} + \frac{b}{b - 24} + \frac{c}{c - 41} \, ? \]

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