
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
Joe the teacher is bad at rounding. Because of this, he has come up with his own way to round grades, where a *grade* is a nonnegative decimal number with finitely many digits after the decimal point.

Given a grade with digits  $a_1a_2 \dots a_m.b_1b_2 \dots b_n$ , Joe first rounds the number to the nearest  $10^{-n+1}$ th place. He then repeats the procedure on the new number, rounding to the nearest  $10^{-n+2}$ th, then rounding the result to the nearest  $10^{-n+3}$ th, and so on, until he obtains an integer.  For example, he rounds the number  $2014.456$  via  $2014.456 \to 2014.46 \to 2014.5 \to 2015$ .

There exists a rational number  $M$  such that a grade  $x$  gets rounded to at least  $90$  if and only if  $x \ge M$ .  If  $M = \tfrac pq$  for relatively prime integers  $p$  and  $q$ , compute  $p+q$ .

*Proposed by Yang Liu*

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