
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
Find the smallest positive integer  $k$  such that
\[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\]
for all positive integers  $b$  and  $x$ . (*Note:* For integers  $a,b,c$  we say  $a \equiv b \pmod c$  if and only if  $a-b$  is divisible by  $c$ .)

*Alex Zhu.*

<details><summary>Clarifications</summary>[list=1][*] ${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$  for all integers  $y$ . In particular,  ${{y}\choose{12}} = 0$  for  $y=1,2,\ldots,11$ .[/list]</details>

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