
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
Define a "ternary sequence" is a sequence that every number is  $0,1$  or  $2$ . ternary sequence  $(x_1,x_2,x_3,\cdots,x_n)$ , define its difference to be   $$ (|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|) $$   A difference will make the length of the sequence decrease by  $1$ , so we define the "feature value" of a ternary sequence with length  $n$  is the number left after  $n-1$  differences. How many ternary sequences has length  $2023$  and feature value  $0$ ?

*Proposed by CSJL*

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