
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 a circle \(C: x^2 + y^2 - 2x - 4y = 0\) and a line \(L: x - y + a = 0\), if the distance from the center of circle \(C\) to line \(L\) is \(\frac{\sqrt{2}}{2}\), determine the value of \(a\).

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