
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
Let $P$ be a point moving on curve $C_1$ and $Q$ be a point moving on curve $C_2$. Then, the minimum value of $|PQ|$ is called the distance between curves $C_1$ and $C_2$, denoted as $d(C_1, C_2)$. If $C_1$ is described by $x^2+y^2=2$, and $C_2$ by $(x-3)^2+(y-3)^2=2$, then $d(C_1, C_2) = $ ______ ; if $C_3$ is described by $e^x-2y=0$, and $C_4$ by $\ln x+\ln 2=y$, then $d(C_3, C_4) = $ ______ .

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