CF 103401E - Power tower

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Solve time: 29s
Verified: no

Solution

I can’t produce a correct editorial yet because the statement that was retrieved for 103401E - Power tower is incomplete and inconsistent across sources. What we currently have is only a truncated and partially duplicated description, and it does not define a coherent computational task (the visible parts describe a sequence $a_n = 2^{a_{n-1}}$ and then suddenly ask for a “limit array” $b_n$, but the definition of $b_n$ and what input $p$ represents is missing or corrupted in the archive page).

From the snippet we can see:

A sequence $a_1 = 2$, $a_n = 2^{a_{n-1}}$.
A statement about a derived array $b_n$, but its definition is not properly included.
Input is $T \le 10^3$, and each query has an integer $p \le 10^7$.
Sample output suggests answers like 0, 1, 4, which clearly implies some number-theoretic or iterated exponent/mod behavior rather than a literal limit of $a_n$ (since that diverges).

However, the key missing piece is the actual definition of what is being computed for each $p$. Without that, any editorial would be fabricated.

What I need from you

Please paste the full problem statement (especially the definition of $b_n$ and what the query integer $p$ represents). Once that is available, I can produce the complete Codeforces-style editorial exactly in your required format, including:

Correct mathematical reformulation of the “power tower”
The actual convergence argument (likely involving modular exponent cycles or fixed points)
Full derivation of the answer formula per $p$
Clean Python solution
Worked traces matching samples

Why this matters

This kind of “power tower limit” problem usually hides a very specific structure (for example:

iterated exponent mod stabilization,
Carmichael function / Euler totient lifting,
fixed points of $2^x$ under some transformation,
or eventual periodicity in exponentiation chains),

and the solution depends entirely on how $b_n$ is defined. Even a small missing clause changes the entire problem class.

Send the full statement, and I’ll turn it into a complete, structured editorial immediately.