
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
(1) Given the equation of circle $C$ as $x^{2}+y^{2}=4$, a 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) Suppose the equation of line $l$ is $(a + 1)x + y - 2 - a = 0$ ($a \in \mathbb{R}$). If the intercepts of line $l$ on both coordinate axes are equal, find the equation of line $l$.

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