
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 $$\vec{a} = (1,1)$$ and $$\vec{b} = (2,3)$$, for what value(s) of \(k\) will the following be true:

(1) The vector \(k\vec{a} + 2\vec{b}\) is perpendicular to \(2\vec{a} - 4\vec{b}\)?
(2) The vector \(k\vec{a} + 2\vec{b}\) is parallel to \(2\vec{a} - 4\vec{b}\)? When they are parallel, are they in the same or opposite direction?

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