
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
Find the maximum value of the positive real number \( k \) such that for any positive real numbers \( a \) and \( b \), the following inequality holds:
$$
\sqrt{a^{2}+k b^{2}}+\sqrt{b^{2}+k a^{2}} \geq a+b+(k-1) \sqrt{a b}
$$

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