
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
Joey wrote a system of equations on a blackboard, where each of the equations was of the form \( a + b = c \) or \( a \cdot b = c \) for some variables or integers \( a, b, c \). Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:

\[
\begin{array}{ll}
x & z = 15 \\
x & y = 12 \\
x & x = 36
\end{array}
\]

If \( x, y, z \) are integer solutions to the original system, find the sum of all possible values of \( 100x + 10y + z \).

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