
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 cube with edge length  $3$  is divided into  $27$  unit cubes. The numbers  $1, 2,\ldots ,27$  are distributed arbitrarily over the unit cubes, with one number in each cube. We form the  $27$  possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). At most how many of the  $27$  row sums can be odd?

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