
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 $\overrightarrow{e_{1}}$, $\overrightarrow{e_{2}}$, and $\overrightarrow{e_{3}}$ be unit vectors, and $\overrightarrow{e_{3}} = \frac{1}{2}\overrightarrow{e_{1}} + k\overrightarrow{e_{2}}$ (where $k > 0$). If the area of the triangle formed by vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ as two sides is $\frac{1}{2}$, then the value of $k$ is \_\_\_\_\_\_.

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