
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 an integer  $n$ , let  $f_9(n)$  denote the number of positive integers  $d\leq 9$  dividing  $n$ .  Suppose that  $m$  is a positive integer and  $b_1,b_2,\ldots,b_m$  are real numbers such that  $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$  for all  $n>m$ .  Find the smallest possible value of  $m$ .

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