
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 $| \overrightarrow{a}|=3$, $| \overrightarrow{b}|=5$, and $| \overrightarrow{a}+ \overrightarrow{b}|=7$.
1. Find the angle $\theta$ between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.
2. Find the value of the real number $k$ such that the vector $k \overrightarrow{a}+ \overrightarrow{b}$ is perpendicular to the vector $\overrightarrow{a}-2 \overrightarrow{b}$.

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