
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
Positive integers a, b, c, d, and e satisfy the equations  $$ (a + 1)(3bc + 1) = d + 3e + 1 $$    $$ (b + 1)(3ca + 1) = 3d + e + 13 $$    $$ (c + 1)(3ab + 1) = 4(26-d- e) - 1 $$  Find  $d^2+e^2$ .


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