
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
Let \( [x] \) represent the greatest integer less than or equal to the real number \( x \), and \( \{x\} = x - [x] \). Given \( a = \frac{p}{q} \) where \( p \) and \( q \) are positive coprime integers, find the value of \( p + q \) given that the sum of all real numbers \( x \) that satisfy \( [x]\{x\} = a x^{2} \) is 420.

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