
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
A chessboard’s squares are labeled with numbers as follows:

[asy]
unitsize(0.8 cm);

int i, j;

for (i = 0; i <= 8; ++i) {
  draw((i,0)--(i,8));
  draw((0,i)--(8,i));
}

for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
  label("$\frac{1}{" + string(9 - i + j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]

Eight of the squares are chosen such that each row and each column has exactly one selected square. Find the maximum sum of the labels of these eight chosen squares.

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