
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 inequality \( ab + b^2 + c^2 \geq \lambda(a + b)c \) holds for all positive real numbers \( a, b, c \) that satisfy \( b + c \geq a \), then the maximum value of the real number \( \lambda \) is \(\quad\) .

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