
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
1) 
\[
\begin{cases}
x = k \sin t + \sin k t \\
y = k \cos t + \cos k t
\end{cases}
\]

Find \(\left(\frac{d y}{d x}\right)_{t=0} \).

2) 
\[
\begin{cases}
x = \alpha^2 + 2 \alpha \\
y = \ln (\alpha+1)
\end{cases}
\]

Find \(\frac{d^2 y}{d x^2} \).

3) 
\[
\begin{cases}
x = 1 + e^{a \varphi} \\
y = a \varphi + e^{-a \varphi}
\end{cases}
\]

Find \(\frac{d^3 y}{d x^3} \).

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