
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

We toss a fair coin \( n \) times. Suppose heads come up \( m \) times. The ratio \( \frac{m}{n} \) is called the frequency of heads. The value \( \frac{m}{n} - 0.5 \) is called the deviation of the frequency from the probability, and the value \( \left| \frac{m}{n} - 0.5 \right| \) is called the absolute deviation. Note that both the deviation and the absolute deviation are random variables. For example, if the coin is tossed 5 times and heads come up twice, the deviation is \( \frac{2}{5} - 0.5 = -0.1 \), and the absolute deviation is 0.1.

The experiment has two parts: first, the coin is tossed 10 times, then it is tossed 100 times. In which of these cases is the expected value of the absolute deviation of the frequency of heads from the probability larger?

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