
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
A grasshopper is sitting on the number line at the number 0. Each second, it jumps one unit to the left or to the right, with equal probability. The directions of the jumps are chosen independently of each other. Let \( p \) denote the probability that, after 2022 jumps, the grasshopper is sitting at a number divisible by 5. Determine whether \( p < \frac{1}{5} \), \( p = \frac{1}{5} \), or \( p > \frac{1}{5} \).

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