
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 average of \( p, q, r \) is 12. The average of \( p, q, r, t, 2t \) is 15. Find \( t \).

\( k \) is a real number such that \( k^{4} + \frac{1}{k^{4}} = t + 1 \), and \( s = k^{2} + \frac{1}{k^{2}} \). Find \( s \).

\( M \) and \( N \) are the points \( (1, 2) \) and \( (11, 7) \) respectively. \( P(a, b) \) is a point on \( MN \) such that \( MP:PN = 1:s \). Find \( a \).

If the curve \( y = ax^2 + 12x + c \) touches the \( x \)-axis, find \( c \).

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