
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 \( a \) be a prime number, and \( b \) be a positive integer such that
\[ 
9(2a + b)^2 = 509(4a + 511b).
\]
Find the values of \( a \) and \( b \).

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