
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 a positive integer  $n$ , let  $f(n)$  be the sum of the positive integers that divide at least one of the nonzero base  $10$  digits of  $n$ . For example,  $f(96)=1+2+3+6+9=21$ . Find the largest positive integer  $n$  such that for all positive integers  $k$ , there is some positive integer  $a$  such that  $f^k(a)=n$ , where  $f^k(a)$  denotes  $f$  applied  $k$  times to  $a$ .


*2021 CCA Math Bonanza Lightning Round #4.3*

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