
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 $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are two non-collinear unit vectors, $\overrightarrow {a}= \overrightarrow {e_{1}}-2 \overrightarrow {e_{2}}$, $\overrightarrow {b}=k \overrightarrow {e_{1}}+ \overrightarrow {e_{2}}$, if $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of $k$; if for any $k \in \mathbb{R}$, $\overrightarrow {a}$ and $\overrightarrow {b}$ can never be perpendicular, find the projection of $\overrightarrow {e_{1}}$ on $\overrightarrow {e_{2}}$.

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