
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 ancient China, the years were recorded using the ten heavenly stems and twelve earthly branches. The ten heavenly stems are: 甲 (jiǎ), 乙 (yǐ), 丙 (bǐng), 丁 (dīng), 戊 (wù), 己 (jǐ), 庚 (gēng), 辛 (xīn), 壬 (rén), 癸 (guǐ); the twelve earthly branches are: 子 (zǐ), 丑 (chǒu), 寅 (yín), 卯 (mǎo), 辰 (chén), 巳 (sì), 午 (wǔ), 未 (wèi), 申 (shēn), 酉 (yǒu), 戌 (xū), 亥 (hài). The ten heavenly stems and twelve earthly branches are arranged in a cyclic manner as follows:<br/>甲乙丙丁戊己庚辛壬癸甲乙丙丁戊己庚辛壬癸$\ldots $<br/>子丑寅卯辰巳午未申酉戌亥子丑寅卯辰巳午未申酉戌亥$\ldots $<br/>From left to right, the first column is 甲子, representing the year of 甲子, the fourth column is 丁卯, representing the year of 丁卯$\ldots $<br/>$(1)$ In the arrangement of the ten heavenly stems above, if 丙 (bǐng) appears for the $n$th time, it is located in the ______ column from left to right (expressed as a formula involving $n$);<br/>$(2)$ The year 2023 is the year of 癸卯 (guǐ mǎo), 癸卯 can be located in the ______ column from left to right (provide one answer).

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