
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 a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) defined for all real numbers \( x \), such that the inequality

$$
\sqrt{2 f(x)} - \sqrt{2 f(x) - f(2x)} \geq 2
$$

holds for all \( x \), the function \( f \) is said to have property \( P \). Find the largest real number \( \lambda \) such that if the function \( f \) has property \( P \), then \( f(x) \geq \lambda \) holds for all real numbers \( x \).

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