
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 the line $y = x + b$ intersect the ellipse $\dfrac{x^2}{2} + y^2 = 1$ at two distinct points $A$ and $B$.

(1) Determine the range of the real number $b$.

(2) Calculate the length of the vector $\left| \overrightarrow{AB} \right|$ when $b=1$.

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