
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
We say a positive integer \( n \) is lovely if there exist a positive integer \( k \) and (not necessarily distinct) positive integers \( d_{1}, d_{2}, \ldots, d_{k} \) such that \( n = d_{1} d_{2} \ldots d_{k} \) and

\[ d_{i}^{2} \mid n + d_{i} \]

for all \( i \in \{1, \ldots, k\} \).

(a) Are there infinitely many lovely numbers?

(b) Does there exist a lovely number greater than 1 which is a square of an integer?

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