
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 $\overrightarrow{e_1}$ and $\overrightarrow{e_2}$ be unit vectors, and the angle between $\overrightarrow{e_1}$ and $\overrightarrow{e_2}$ is $\frac{\pi}{3}$. If $\overrightarrow{a}= \overrightarrow{e_1}+3 \overrightarrow{e_2}$ and $\overrightarrow{b}=2 \overrightarrow{e_1}$, then the projection of vector $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$ is ______.

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