
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
Let  $\mathcal{P}$  be the set of vectors defined by
\[\mathcal{P} = \left\{\begin{pmatrix} a  b \end{pmatrix} \, \middle\vert \, 0 \le a \le 2, 0 \le b \le 100, \, \text{and} \, a, b \in \mathbb{Z}\right\}.\]
Find all  $\mathbf{v} \in \mathcal{P}$  such that the set  $\mathcal{P}\setminus\{\mathbf{v}\}$  obtained by omitting vector  $\mathbf{v}$  from  $\mathcal{P}$  can be partitioned into two sets of equal size and equal sum.

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