
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
If \( a = -1 + 2 - 3 + 4 - 5 + 6 - \ldots + 100 \), find \( a \).

The sum of the first \( b \) positive odd numbers is \( 2a \). Find \( b \).

A bag contains \( b \) white balls and 3 black balls. Two balls are drawn from the bag at random. If the probability of getting 2 balls of different colors is \( \frac{c}{13} \), find \( c \).

If the lines \( cx + 10y = 4 \) and \( dx - y = 5 \) are perpendicular to each other, find \( d \).

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