
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
Let the sequence  $\{x_n\}$  be defined by  $x_1 \in \{5, 7\}$  and, for  $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$ . For example, the possible values of  $x_3$  are  $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$ , and  $7^{7^7}$ . Determine the sum of all possible values for the last two digits of  $x_{2012}$ .

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