
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
The sum of 2 numbers is 20, their product is 10. If the sum of their reciprocals is $a$, find $a$.

$21^{2}-1=0 \times 2, 2^{2}-1=1 \times 3, 3^{2}-1=2 \times 4, \ldots, b^{2}-1=135 \times 137$. If $b>0$, find $b$.

If the lines $x+2y+1=0$ and $cx+3y+1=0$ are perpendicular, find $c$.

The points $(2,-1), (0,1), (c, d)$ are collinear. Find $d$.

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