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Garden Click Coefficient

Reference time limit: 0.5 s; memory limit: 128 MiB. Use twice the slowest formal-file median on the target platform, rounded up to 0.5 seconds. See the benchmark environment.

Story

A cookie factory has unlocked a small garden. Its gardener may change the plants and soil to maximize the cookie coefficient obtained from each click when the garden is unfrozen.

(This story was rewritten by GPT-6.)

Formal Problem Statement

Each cell of an \(n\times m\) rectangle has exactly one type from the table. All cells use the same soil coefficient \(S\in\{1,1.25\}\) . The achievement count is \(A\) .

Type	Meaning	Effect
E	Empty	None
B	Wheat	Production bonus
T	Corn	Click bonus
W	Bloom	Milk bonus
N	Nurse tulip	Amplifies neighboring plants
D	Fern	Production bonus and click penalty
G	Glove mushroom	Click and cursor bonuses, production penalty

Two distinct cells are neighbors when both coordinate differences are at most one. If cell \(i\) has \(a_i\) neighboring N cells, its effect coefficient, when it is not N, is

\[ F_i=S(1+0.2a_iS). \]

N cells do not amplify other N cells and have no direct production effect. A type letter below denotes the set of cells of that type. Empty sums are zero and empty products are one.

\[ \begin{aligned} C_{\rm cursor}&=1+0.01\sum_{i\in G}F_i,\\ C_{\rm production}&=1+0.01\sum_{i\in B}F_i+0.03\sum_{i\in D}F_i,\\ D_{\rm production}&=\prod_{i\in G}(1-0.01F_i),\\ C_{\rm click}&=0.02\sum_{i\in T}F_i+0.04\sum_{i\in G}F_i,\\ D_{\rm click}&=\prod_{i\in D}(1-0.05F_i),\\ f_W&=0.002\sum_{i\in W}F_i,\\ M&=0.04A\cdot1.05^2\cdot1.1\cdot f_W. \end{aligned} \]

\[ K=(1+0.05M)(1+0.1M)^2(1+0.115M)(1+0.125M)^2 (1+0.15M)^2(1+0.175M)^2(1+0.2M)^5. \]

Maximize

\[ C=(0.956+0.044C_{\rm cursor})K C_{\rm production}D_{\rm production}C_{\rm click}D_{\rm click}. \]

These formulas fully define the problem. Neither \(C_{\rm click}\) nor \(f_W\) has an additional constant one. Ignore golden-cookie effects.

The N cell affects every neighboring non-N cell, including diagonal neighbors.

Input Format

One line contains n m A .

Output Format

Print the maximum coefficient. The absolute or relative error may be at most \(10^{-9}\) : an output \(x\) and optimum \(y\) must satisfy \(|x-y|\le10^{-9}\max(1,|y|)\) .

Example 1

Input

1 1 0

Output

4.940215625000000000e-02

Example 2

Input

2 2 1000

Output

1.906043514560546875e-01

Example 3

Input

2 4 1000

Output

5.854567874453084032e-01

Constraints and Guarantees

\(n,m\ge1\) , \(nm\le8\) , and \(0\le A\le1000\) .

Redesign by Codex (GPT-6): the source omitted bounds, output rules, and examples. This edition supplies them and resolves the conflicting neighbor formula in favor of the stated 20% amplification, using \(0.2a_iS\) . All other factors and exponents are retained.

Score	Additional conditions
100%	None

Ten formal files, worth 10% each. Generation guide.