
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
Let $\vec{i}$ and $\vec{j}$ represent the unit vectors along the $x$ and $y$ axes in a plane rectangular coordinate system, respectively. Given that $|\vec{a} - \vec{i}| + |\vec{a} - 2\vec{j}| = \sqrt{5}$, determine the possible range of values for $|\vec{a} + 2\vec{i}|$.

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