
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 non-collinear unit vectors $\overrightarrow{e_{1}}$, $\overrightarrow{e_{2}}$:

(1) If the angle between $\overrightarrow{e_{1}}$, $\overrightarrow{e_{2}}$ is $\frac{\pi}{3}$, find the magnitude of the vector $\overrightarrow{e_{1}}-2\overrightarrow{e_{2}}$.

(2) If the angle between $\overrightarrow{e_{1}}$, $\overrightarrow{e_{2}}$ is $\frac{\pi}{4}$, for a non-zero vector $\overrightarrow{a}=x\overrightarrow{e_{1}}+y\overrightarrow{e_{2}}$, where $x,y\in R$, find the maximum value of $\frac{|x|}{|\overrightarrow{a}|}$.

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