
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*}
  13x +  by +  cz &= 0 
   ax + 23y +  cz &= 0 
   ax +  by + 42z &= 0.
\end{align*}Suppose that  $ a \ne 13$  and  $ x \ne 0$ .  What is the value of
\[ \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} \, ?\]

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