
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
suppose that  $a=3^{100}$  and  $b=5454$ . how many  $z$ s in  $[1,3^{99})$  exist such that for every  $c$  that  $gcd(c,3)=1$ , two equations  $x^z\equiv c$  and  $x^b\equiv c$  (mod  $a$ ) have the same number of answers?( $\frac{100}{6}$  points)

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