
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
Definition: The minimum value of the distance from a point on curve $C$ to line $l$ is called the distance from curve $C$ to 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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