
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
\[
   17x +  by +  cz = 0, \\
   ax + 29y +  cz = 0, \\
   ax +  by + 53z = 0.
\]
Suppose that \( a \ne 17\) and \( x \ne 0\). What is 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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