
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 every real number  $ a, $  define the set  $ A_a=\left\{ n\in\{ 0\}\cup\mathbb{N}\bigg|\sqrt{n^2+an}\in\{ 0\}\cup\mathbb{N}\right\} . $ **a)** Show the equivalence:  $ \# A_a\in\mathbb{N}\iff a\neq 0,  $  where  $ \# B $  is the cardinal of  $ B. $ **b)** Determine  $ \max A_{40} . $ 

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