
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, z$ be real numbers that satisfy the three equations
\[
  14x + by + cz = 0, \\
  ax + 24y + cz = 0, \\
  ax + by + 43z = 0.
\]
Suppose that $a \ne 14, b \ne 24, c \ne 43$, and $x \ne 0$. What is the value of
\[
\frac{a}{a - 14} + \frac{b}{b - 24} + \frac{c}{c - 43} \, ?
\]

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