
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 the straight line $y=x+b$ intersects with the curve $({x-2})^{2}+({y-3})^{2}=4 (0 \leqslant x \leqslant 4, 1 \leqslant y \leqslant 3)$, determine the range of the real number $b$.

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