
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 the circle C: $x^2 + y^2 = 4$.
(1) Line $l$ passes through point $P(1, 2)$ and intersects circle $C$ at points $A$ and $B$. If $|AB| = 2\sqrt{3}$, find the equation of line $l$.
(2) For a point $M$ moving on the circle $C$, construct a line $m$ parallel to the y-axis through $M$. Let point $N$ be the intersection of $m$ with the x-axis. If vector $\overrightarrow{OQ} = \overrightarrow{OM} + \overrightarrow{ON}$, find the equation of the trajectory of point $Q$.
(3) If point $R$ is at $(1, 0)$, under the conditions of (2), find the minimum value of $|\overrightarrow{RQ}|$.

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