
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
For each natural number $n$, let $f(n)$ represent the number of non-negative integer solutions $(x, y, z)$ to the indeterminate equation $x + 2y + 5z = n$. Determine all $n$ such that $5f(n)$ is a perfect square, and find the corresponding square expression.

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