
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 two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with an angle of $120^\circ$ between them, $|\overrightarrow {a}|=1$, and $|\overrightarrow {b}|=3$, find the length of the vector $|5 \overrightarrow {a}- \overrightarrow {b}|$, and the projection of the vector $2 \overrightarrow {a}+ \overrightarrow {b}$ in the direction of $\overrightarrow {b}$.

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