
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 $O$ be the coordinate origin. If line $l$: $y=x+b (b > 0)$ intersects the circle $x^{2}+y^{2}=4$ at two distinct points $P_{1}$, $P_{2}$, and $| \overrightarrow{P_{1}P_{2}}| \geqslant | \overrightarrow{OP_{1}} + \overrightarrow{OP_{2}}|$, calculate the maximum value of the real number $b$.

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