
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
If vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ satisfy: $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, and when $\lambda \in \mathbb{R}$, the minimum value of $|\overrightarrow{b}-\lambda \overrightarrow{a}|$ is $2\sqrt{2}$, calculate the projection of vector $\overrightarrow{a} + \overrightarrow{b}$ in the direction of vector $\overrightarrow{a}$.

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