
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
Indicate the number of steps sufficient to assert, with a probability of error not exceeding 0.001, that the resulting speed of the token is less than 0.01.

Recall that each movement of the token is determined by the result of a coin toss. If, in $n$ coin tosses, heads appears $l$ times and tails $n-l$ times, the token will make $l$ steps to the right and $n-l$ steps to the left, ending up at the point

$$
l - (n - l) = 2l - n
$$

The resulting speed of the token after $n$ steps is given by the absolute value of the ratio:

$$
\frac{2l - n}{n} = 2 \frac{l}{n} - 1
$$

The fraction $\frac{l}{n}$ represents the frequency of heads.

Choose an acceptable probability of error $\varepsilon$ and let $a$ be an arbitrarily small number. If the number of coin tosses exceeds

$$
N = \frac{1}{a^2 \sqrt[3]{\varepsilon^2}}
$$

then with a probability of error less than $\varepsilon$, the frequency of heads differs from $\frac{1}{2}$ by less than $a$.

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