
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 satisfying the equations:
\begin{align*}
  17x + by + cz &= 0 \\
  ax + 29y + cz &= 0 \\
  ax + by + 53z &= 0.
\end{align*}
Assume that $a \ne 17$ and $x \ne 0$. Determine the value of
\[
\frac{a}{a - 17} + \frac{b}{b - 29} + \frac{c}{c - 53}.
\]

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