
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
The numbers  $1$  through  $9$  are written on a  $3 \times 3$  board, without repetitions. A valid operation is to choose a row or a column of the board, and replace its three numbers  $a, b, c$  (in order, i.e., the first number of the row/column is  $a$ , the second number of the row/column is  $b$ , the third number of the row/column is  $c$ ) with either the three non-negative numbers  $a-x, b-x, c+x$  (in order) or with the three non-negative numbers  $a+x, b-x, c-x$  (in order), where  $x$  is a real positive number which may vary in each operation .

a) Determine if there is a way of getting all  $9$  numbers on the board to be the same, starting with the following board: $\begin{array}{|c|c|c|c|c|c|c|c|} \hline 1 & 2 & 3  \hline 4 & 5 & 6  \hline 7 & 8 & 9  \hline \end{array}$ b) For all posible configurations such that it is possible to get all  $9$  numbers to be equal to a number  $m$  using the valid operations, determine the maximum value of  $m$ .

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