
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
In the Cartesian coordinate system, Xiaoming plays a chess game where the piece starts at the origin. The rules are as follows: on the $1$st step, move $1$ unit to the right; on the $2$nd step, move $2$ units to the right; on the $3$rd step, move $1$ unit up; on the $4$th step, move $1$ unit to the right $\ldots$ and so on. For the $n$th step: when $n$ is divisible by $3$, move $1$ unit up; when the remainder of $n$ divided by $3$ is $1$, move $1$ unit to the right; when the remainder of $n$ divided by $3$ is $2$, move $2$ units to the right. After the $100$th step, the coordinates of the piece are ____.

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