
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
Define the minimum distance from a point on curve C to line l as the distance between curve C and line l. Given that the distance from curve $C_1: y = x^2 + a$ to line $l: y = x$ is equal to the distance from curve $C_2: x^2 + (y+4)^2 = 2$ to line $l: y = x$, find the value of the real number $a$.

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