
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
If  $ a,n $  are two natural numbers corelated by the equation  $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $  then  $ 
1+4a $  is a perfect square. Justify this statement. Here,  $ \{\} $  is the usual fractionary part.

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