
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
An up-right path from lattice points  $P$  and  $Q$  on the  $xy$ -plane is a path in which every move is either one unit right or one unit up. The probability that a randomly chosen up-right path from  $(0,0)$  to  $(10,3)$  does not intersect the graph of  $y=x^2+0.5$  can be written as  $\tfrac mn$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by **HrishiP***

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