
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 real numbers $a$, $b$, $c$ satisfying $a \leq b \leq c$ and $ab + bc + ca = 0$, with $abc = 1$. Find the largest real number $k$ such that the inequality $|a + b| \geq k|c|$ always holds.

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