
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 $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=1$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $45^{\circ}$.  
$(1)$ Find the projection of $\overrightarrow{a}$ onto the direction of $\overrightarrow{b}$.  
$(2)$ Find the value of $|\overrightarrow{a}+2\overrightarrow{b}|$.  
$(3)$ If the angle between the vectors $(2\overrightarrow{a}-\lambda \overrightarrow{b})$ and $(\lambda \overrightarrow{a}-3\overrightarrow{b})$ is acute, find the range of real number $\lambda$.

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