
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 sequence  $(x_{n})_{n\geq 1}$  by taking  $x_{1}\in\left\{5,7\right\}$ ; when  $k\ge 1$ ,  $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$ . Determine all possible last two digits of  $x_{2009}$ .

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