
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
The line \( x = k \) intersects the curves \( y = \log_{5} x \) and \( y = \log_{5} (x+4) \) at two points, and the distance between these points is 0.5. Given \( k = a + \sqrt{b} \), where \( a \) and \( b \) are integers, find the value of \( a + b \).

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