
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 unit vectors, $\overrightarrow {b}$ is a non-zero vector such that $\overrightarrow {b}$=x $\overrightarrow {e_{1}}$+y $\overrightarrow {e_{2}}$, where x, y belong to the set of real numbers. If the angle between $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ is $\frac {π}{6}$, find the maximum value of $\frac {|x|}{| \overrightarrow {b}|}$.

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