
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 point  $P$  on a real number line is moved according to the rule as follows.
Flip a coin, if heads is the spot on the face, then  $P$  is moved by  $+1$ , and if tails is the spot on the face, then  $P$  is moved to the point of reflection for the origin. At first  $P$  is set on the origin on the real number, let  $a_n$  be the coordinate of  $P$  after  $n$  tossing the coin.

(1) What's the probability such that  $a_3=0?$ (2) What's the probability such that  $a_4=1?$ (3) Express the probability such that  $a_n=n-3$  for  $n\geq 3$  in terms of  $n$ .

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