
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 points $A(-1,2)$ and $B(0,1)$, and a moving point $P$ that satisfies $|PA| = \sqrt{2}|PB|$.

(I) If the trajectory of point $P$ forms a curve $C$, find the equation of this curve.

(II) If point $Q$ is on the line ${l_1}: 3x - 4y + 12 = 0$, and line ${l_2}$ passes through point $Q$ and intersects curve $C$ at exactly one point $M$, find the minimum value of $|QM|$.

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