
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 vectors $\overrightarrow{a}=(1,0)$ and $\overrightarrow{b}=(m,1)$, with an angle of $\frac{\pi}{4}$ between them.

(1) Find $| \overrightarrow{a}-2\overrightarrow{b} |$;

(2) If $(\overrightarrow{a}+\lambda\overrightarrow{b})$ is perpendicular to $\overrightarrow{b}$, find the value of the real number $\lambda$.

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