
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
R is the reals. X is the square \([0, 1] \times [0, 1]\). \(f : X \rightarrow R\) is continuous. If \(\int_{Y} f(x, y) \, dx \, dy = 0\) for all squares \(Y\) such that:
1. \(Y \subseteq X\),
2. \(Y\) has sides parallel to those of \(X\),
3. at least one of \(Y\)'s sides is contained in the boundary of \(X\),

is it true that \(f(x, y) = 0\) for all \(x, y\)?

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