
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, and $|\overrightarrow{{e}_{1}}+2\overrightarrow{{e}_{2}}|\leq2$, if a non-zero vector $\overrightarrow{a}$ satisfies $\overrightarrow{a}•\overrightarrow{{e}_{1}}\leq\overrightarrow{a}•\overrightarrow{{e}_{2}}$, calculate the maximum value of $\frac{\overrightarrow{a}•(2\overrightarrow{{e}_{1}}+\overrightarrow{{e}_{2}})}{|\overrightarrow{a}|}$.

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