
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
Given that $a$ is a positive integer and $a = b - 2005$, if the equation $x^2 - ax + b = 0$ has a positive integer solution, what is the minimum value of $a$?  
(Hint: First, assume the two roots of the equation are $x_1$ and $x_2$, then…)

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