
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 that $x, y \in \mathbb{R}$, vectors $\overset{→}{a}=(x,1,0)$, $\overset{→}{b}=(1,y,0)$, and $\overset{→}{c}=(2,-4,0)$ satisfy $\overset{→}{a} \bot \overset{→}{c}$ and $\overset{→}{b} \parallel \overset{→}{c}$, find the value of $|\overset{→}{a}+\overset{→}{b}|$.

A) $\sqrt{5}$
B) $\sqrt{10}$
C) $2\sqrt{5}$
D) $10$

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