
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 the line $y=x+2a$ intersects the circle $C:x^{2}+y^{2}-2ay-2=0\left(a \gt 0\right)$ at points $A$ and $B$, and $|AB|=2\sqrt{3}$, find the value of $a$.

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