
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 four distinct points on a plane: $A(1, -2a)$, $B(2, a)$, $C(2+a, 0)$, and $D(2a, 1)$.

(1) For what value of $a$ are points $A$, $B$, and $C$ collinear?

(2) For what value of $a$ are lines $AB$ and $CD$ perpendicular?

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