
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
Solve the equations:
1) \( y^{\prime \prime}-5 y^{\prime}-6 y=0 \)
2) \( y^{\prime \prime \prime}-6 y^{\prime \prime}+13 y^{\prime}=0 \)
3) \( \frac{d^{2} S}{d t^{2}}+4 \frac{d S}{d t}+4 S=0 \)
4) \( \frac{d^{4} y}{d x^{4}}-y=0 \)
5) \( y^{(4)}+13 y^{(2)}+36 y=0 \)
6) \( y^{(2)}+2 y^{(5)}+y^{(3)}=0 \)

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