
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
For each pair of positive integers  $(x, y)$  a nonnegative integer  $x\Delta y$  is defined. 
It is known that for all positive integers  $a$  and  $b$  the following equalities hold:
i.  $(a + b)\Delta b = a\Delta b + 1$ .
ii.  $(a\Delta b) \cdot  (b\Delta a) = 0$ .
Find the values of the expressions  $2016\Delta 121$  and  $2016\Delta 144$ .

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