
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
Let $x, y \in \mathbb{R}$, and vectors $\vec{a} = (x, 2)$, $\vec{b} = (1, y)$, and $\vec{c} = (2, -6)$. Given that $\vec{a}$ is perpendicular to $\vec{c}$ and $\vec{b}$ is parallel to $\vec{c}$, find the magnitude of the vector $\vec{a} + \vec{b}$.

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