
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
On the planet Lemniscate, the people use an advanced version of the periodic table. They know that $\mathrm{Kr}$ deteriorates into $\mathrm{Pl}$ and then into a stable element As. Starting with a block of $\mathrm{Kr}$ of size 10, at time $t$, let:
- $x(t)$ be the amount of $\mathrm{Kr}$,
- $y(t)$ be the amount of $\mathrm{Pl}$,
- $z(t)$ be the amount of As.

Given $x^{\prime}(t)=-x$, and in the absence of $\mathrm{Kr}$, $y^{\prime}(t)=-2y$, find the time $t$ at which the quantity of $\mathrm{Pl}$ will be the largest. Assume the entire amount of deteriorating $\mathrm{Kr}$ turns into $\mathrm{Pl}$.

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