
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  $a=1+10^{-4}$ . Consider some  $2023\times 2023$  matrix with each entry a real in  $[1,a]$ . Let  $x_i$  be the sum of the elements of the  $i$ -th row and  $y_i$  be the sum of the elements of the  $i$ -th column for each integer  $i\in [1,n]$ . Find the maximum possible value of  $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$  (the answer may be expressed in terms of  $a$ ).

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