
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

Solve the following system of equations:

$$
\begin{aligned}
& 52x - 34y - z + 9a = 0 \\
& 49x - 31y - 3z - b = 0 \\
& 36x - 24y + z + 3a + 2b = 0
\end{aligned}
$$

Given that \( a \) and \( b \) are positive numbers, determine the relation between \( a \) and \( b \) so that all unknowns are positive. Additionally, identify the values of \( a \) and \( b \) for which the value of \( x \) is minimal.

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