
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
Let  $f$  be the quadratic function with leading coefficient  $1$  whose graph is tangent to that of the lines  $y=-5x+6$  and  $y=x-1$ .  The sum of the coefficients of  $f$  is  $\tfrac pq$ , where  $p$  and  $q$  are positive relatively prime integers.  Find  $100p + q$ .

*Proposed by David Altizio*

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