
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  $a$  be a positive real number,  $n$  a positive integer, and define the *power tower*  $a\uparrow n$  recursively with  $a\uparrow 1=a$ , and  $a\uparrow(i+1)=a^{a\uparrow i}$  for  $i=1,2,3,\ldots$ .  For example, we have  $4\uparrow 3=4^{(4^4)}=4^{256}$ , a number which has  $155$  digits.  For each positive integer  $k$ , let  $x_k$  denote the unique positive real number solution of the equation  $x\uparrow k=10\uparrow (k+1)$ .  Which is larger:  $x_{42}$  or  $x_{43}$ ?

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