
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 that plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=1$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $45^{\circ}$.  
$(1)$ Find the value of $|\overrightarrow{a}+2\overrightarrow{b}|$;  
$(2)$ If the vectors $2\overrightarrow{a}-\lambda\overrightarrow{b}$ and $\lambda\overrightarrow{a}-3\overrightarrow{b}$ are parallel, find the value of the real number $\lambda$.

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