
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 vectors $\overrightarrow{a}= \overrightarrow{e_{1}}+2 \overrightarrow{e_{2}}$ and $\overrightarrow{b}=-3 \overrightarrow{e_{1}}+2 \overrightarrow{e_{2}}$, where $\overrightarrow{e_{1}} \perp \overrightarrow{e_{2}}$ and $\overrightarrow{e_{1}}^{2}= \overrightarrow{e_{2}}^{2}=1$:

1. Calculate the value of $|\overrightarrow{a}+ \overrightarrow{b}|$;
2. For what value of $k$ are $k\overrightarrow{a}+ \overrightarrow{b}$ and $\overrightarrow{a}-3\overrightarrow{b}$ perpendicular to each other?

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