
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
If \( a \), \( b \), and \( c \) are distinct integers, then the minimum value of
$$
3a^2 + 2b^2 + 4c^2 - ab - 3bc - 5ca
$$
is $\qquad$.

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