
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 real number \( k \) such that for any positive real numbers \( a, b, c, d \), the following inequality holds:
\[
(a+b+c)\left[3^{4}(a+b+c+d)^{5}+2^{4}(a+b+c+2d)^{5}\right] \geqslant k a b c d^{3}.
\]

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