
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 largest real number \( k \) such that the inequality
\[
(a+b+c)^{3} \geq \frac{5}{2}\left(a^{3}+b^{3}+c^{3}\right)+k a b c
\]
holds for any side lengths \( a \), \( b \), and \( c \) of a triangle.

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