
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 vertical oscillations of a mass $m$ on a spring with stiffness $k$ in a viscous medium are described by the equation $x(t)=a e^{-2 t}+b e^{-t}+m g / k$, where $a$ and $b$ are constants depending on the initial conditions, and $t$ is the time. At the initial moment, the load was taken out of equilibrium. Given that $\mathrm{mg} / \mathrm{k}=1$ and $b=1-2 a$, find the values of the constant $a$ for which the spring will be in an unextended state ( $x=0$ ) twice during the motion.

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